Response Modeling Methodology
This page intentionally left blank
Series on Quality, Reliability and Engineering Sta...
54 downloads
1055 Views
2MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
Response Modeling Methodology
This page intentionally left blank
Series on Quality, Reliability and Engineering Statistics
Response Modeling Methodology Empirical Modeling for Engineering and Science
Haim Shore Ben-Gurian University of the Negev, Israel
We World Scientific NEW JERSEY · LONDON · SINGAPORE · BEIJING · SHANGHAI · HONG KONG · TAIPEI · CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Series on Quality, Reliability and Engineering Statistics — Vol. 8 RESPONSE MODELING METHODOLOGY (RMM) Empirical Modeling for Engineering and Science Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-256-102-1
Printed in Singapore.
'HGLFDWHGWRP\SDUHQWV+DYDDQG'DQLHOZKRVHOLYHV KDYHKHURLFDOO\GHILHGDGYHUVLW\
Y
This page intentionally left blank
3UHIDFH
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
YLL
YLLL
35()$&(
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³DUVHQDO´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
35()$&(
L[
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±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
[
35()$&(
3DUW,90RGHOLQJ5DQGRP9DULDWLRQ$SSOLFDWLRQV /LNH LWV SUHGHFHVVRU WKLV SDUW RI WKH ERRN GHYHORSV QHZ 500EDVHG PRGHOV IRU YDULRXV DUHDV LQ HQJLQHHULQJ DQG RSHUDWLRQVPDQDJHPHQW ,W VWDUWV ZLWK &KDSWHU ZKHUH ZH GHPRQVWUDWH WKH JRRGQHVVRIILW DFKLHYDEOHZKHQWKH500HUURUGLVWULEXWLRQLVXVHGWRPRGHO UHSODFH VRPH FXUUHQW FRPPRQO\ HQFRXQWHUHG VWDWLVWLFDO GLVWULEXWLRQV ,QYHUVH QRUPDOL]LQJ WUDQVIRUPDWLRQV ,17V DQG SLHFHZLVH OLQHDU DSSUR[LPDWLRQV ERWK GHULYDWLYHV RI 500 WKDW PD\ EH XVHIXO LQ PRGHOLQJUDQGRPYDULDWLRQDUHGHYHORSHGDQGGHPRQVWUDWHGLQ&KDSWHUV DQGUHVSHFWLYHO\:HWKHQSURFHHGWRGHPRQVWUDWHDSSOLFDWLRQRI 500 DQG LWV GHULYDWLYHV WR 6WDWLVWLFDO 3URFHVV &RQWURO 63& ZKHQ WUDGLWLRQDO6KHZKDUWFKDUWVFDQQRWEHLPSOHPHQWHG &KDSWHU DQGWR PRGHOLQJDQGRSWLPL]DWLRQLQWKHDUHDRILQYHQWRU\DQDO\VLV &KDSWHU 6RPHUHYLHZTXHVWLRQVDUHSURYLGHGLQWKHODVWFKDSWHU 7KURXJKRXW WKH ERRN ORJ < PHDQV ORJH < >OQ < @ 5HOHYDQW UHIHUHQFHVDUHJLYHQVHSDUDWHO\IRUHDFKFKDSWHUUHQGHULQJWKHERRNPRUH XVHUIULHQGO\500UHODWHGPDWHULDOV OLNHXSGDWHGUHIHUHQFHVOLVWGDWD VHWVDQGDOJRULWKPV PD\EHIRXQGDWWKHDXWKRU VSHUVRQDOVLWHUHDFKDEOH IURP KWWS ZZZLHEJXDFLOLHP 7KH DXWKRU ZLVKHV WR WKDQN 0LVV &KHOVHD &KLQ RI :RUOG 6FLHQWLILF 3XEOLVKLQJ DQG KHU WHDP IRU WKHLU YDOXDEOH DVVLVWDQFH WKURXJKRXW WKH DUGXRXVHGLWLQJSURFHVV7KHLUVKDUHLQREWDLQLQJWKHUHTXLUHGTXDOLW\LV YHU\PXFKDSSUHFLDWHG :H KRSH WKDW WKLV ERRN ZLOO SURYLGH D IUHVK DSSURDFK WR HPSLULFDO PRGHOLQJ LQ HQJLQHHULQJ RSHUDWLRQVPDQDJHPHQW DQG DSSOLHG VFLHQFH ZKHUH FXUUHQW WRROV PD\ EH WRR FRPSOLFDWHG WR LPSOHPHQW UHTXLUH WRR PDQ\WKHRUHWLFDODVVXPSWLRQVWRDSSO\RUGRQRWUHODWHUHVSHFWIXOO\WRWKH VWRU\WKDWWKHGDWDDUHWHOOLQJ +$,06+25( %HHU6KHYD -DQXDU\
5HIHUHQFHV &RPPHQW 7KH UHIHUHQFHV EHORZ DUH DQ XSGDWHG OLVW RI SXEOLFDWLRQV ZKLFKDGGUHVV500DQGLWVGHULYDWLYHV
>@ 6KRUH + D *HQHUDO FRQWURO FKDUWV IRU YDULDEOHV ,QWHUQDWLRQDO -RXUQDO RI 3URGXFWLRQ5HVHDUFK >@6KRUH+ E *HQHUDOFRQWUROFKDUWVIRUDWWULEXWHV,,(7UDQVDFWLRQV 4XDOLW\DQG 5HOLDELOLW\(QJLQHHULQJ >@ 6KRUH + F 7KUHH DSSURDFKHV WR DQDO\]H TXDOLW\ GDWD RULJLQDWLQJ LQ QRQ QRUPDOSRSXODWLRQV4XDOLW\(QJLQHHULQJ >@ 6KRUH + D 0RGHOLQJ D QRQQRUPDO UHVSRQVH IRU TXDOLW\ LPSURYHPHQW ,QWHUQDWLRQDO-RXUQDORI3URGXFWLRQ5HVHDUFK >@ 6KRUH + E ,QYHUVH QRUPDOL]LQJ WUDQVIRUPDWLRQV DQG D QRUPDOL]LQJ WUDQVIRUPDWLRQ ,Q ³$GYDQFHV LQ 0HWKRGRORJLFDO DQG $SSOLHG $VSHFWV RI 3UREDELOLW\ DQG 6WDWLVWLFV´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
[L
This page intentionally left blank
&RQWHQWV
3UHIDFH 5HIHUHQFHV[L ,QWURGXFWLRQ
YLL
5HIHUHQFHV 3$57,&855(1702'(/6$1'02'(/,1* 0(7+2'2/*,(6 5HODWLRQDO0RGHOVLQ(QJLQHHULQJDQGWKH6FLHQFHV 0RQRWRQH&RQYH[&RQFDYH5HODWLRQVKLSV
,QWURGXFWLRQ &KHPLVWU\DQG&KHPLFDO(QJLQHHULQJ 3K\VLFV (OHFWULFDO(QJLQHHULQJ +DUGZDUH5HOLDELOLW\(QJLQHHULQJ 6RIWZDUH5HOLDELOLW\*URZWK0RGHOLQJ *URZWK0RGHOV 5HIHUHQFHV
6KDUHG)HDWXUHVDQG7KH/DGGHU ,QWURGXFWLRQ 6KDUHG)HDWXUHV
[LLL
[LY
&217(176
7KH/DGGHURI)XQGDPHQWDO8QLIRUPO\&RQYH[&RQFDYH )XQFWLRQV
$SSURDFKHVWR0RGHO6\VWHPDWLF9DULDWLRQ
,QWURGXFWLRQ /LQHDU5HJUHVVLRQ$QDO\VLV %R[&R[3RZHU7UDQVIRUPDWLRQV *HQHUDOL]HG/LQHDU0RGHOV &RQFOXVLRQV 5HIHUHQFHV
$SSURDFKHVWR0RGHO5DQGRP9DULDWLRQ
,QWURGXFWLRQ 3DUDPHWHU5LFK)DPLOLHVRI'LVWULEXWLRQV7UDQVIRUPDWLRQVDQG ([SDQVLRQV 7KH3HDUVRQIDPLO\RIGLVWULEXWLRQV 2WKHUIDPLOLHVRIGLVWULEXWLRQV %XUU7XNH\ VJDQG KV\VWHPVJHQHUDOL]HG/DPEGD6KRUHWKHH[SRQHQWLDO IDPLO\ 7UDQVIRUPDWLRQV -RKQVRQ%R[&R[ DQGH[SDQVLRQV 0RPHQWVDQG7KHLU5ROHLQ(PSLULFDO0RGHOLQJRI D'LVWULEXWLRQ :K\PRPHQWPDWFKLQJ" +RZPDQ\PRPHQWVWRPDWFK +HXULVWLF0HWKRGVLQ(PSLULFDO0RGHOLQJRI5DQGRP 9DULDWLRQ $Q$OWHUQDWLYH$SSURDFKWR)RXU0RPHQW0DWFKLQJ 5HIHUHQFHV
7KH5HTXLUHPHQWVDQG(YDOXDWLRQRI&RPSOLDQFH
,QWURGXFWLRQ 'HVLUDEOH5HTXLUHPHQWVRID*HQHUDO0HWKRGRORJ\IRU(PSLULFDO 0RGHOLQJ $Q(YDOXDWLRQRI&RPSOLDQFHRI&XUUHQW0HWKRGRORJLHV 0RGHOLQJV\VWHPDWLFYDULDWLRQ 0RGHOLQJUDQGRPYDULDWLRQ
&217(176
5HIHUHQFHV 3$57,,500'(9(/23,1*$1'(9$/8$7,1* 7+(*(1(5$/$3352$&+ 7KH5000RGHO
[Y
,QWURGXFWLRQ $Q$[LRPDWLF'HULYDWLRQRIWKH5000RGHO 7KHPRGHODVVXPSWLRQV 7KHJHQHUDOPRGHO 'HULYLQJI 'HULYLQJI 7KH5000RGHO 7KH5HVSRQVH0RPHQWV ([SORULQJWKH5HODWLRQVKLSEHWZHHQWKH&9DQGη 5HIHUHQFHV
(VWLPDWLQJWKH5HODWLRQDO0RGHO
,QWURGXFWLRQ 3KDVH(VWLPDWLQJWKH/LQHDU3UHGLFWRU /3 ,QWURGXFWLRQDQGPRWLYDWLRQ 6WDJH,$SSUR[LPDWLQJDWUDQVIRUPHGUHVSRQVHYLD D7D\ORUVHULHVH[SDQVLRQDQGHVWLPDWLQJWKHSDUDPHWHUV YLD&&$ 6WDJH,,6WHSZLVHOLQHDUUHJUHVVLRQDQDO\VLVZLWK FDQRQLFDOVFRUHVDVUHVSRQVHYDOXHV ,VVXHV5HODWHGWR,PSOHPHQWDWLRQRI3KDVH 3KDVH(VWLPDWLQJWKH5000RGHO ,QWURGXFWLRQ 6WDJH,(VWLPDWLQJWKH500SDUDPHWHUV^α, λ, µ2 ` 6WDJH,,(VWLPDWLQJWKH500(UURU3DUDPHWHUV {ρ, σε1, σε2}, 6XPPDU\RIWKHHVWLPDWLRQSURFHGXUH 3KDVH 7ZR1XPHULFDO([DPSOHV ([DPSOH7KH:DYH6ROGHULQJ3URFHVV ([DPSOH7KH5HVLVWLYLW\'DWD 5HIHUHQFHV
[YL
&217(176
$SSHQGL[$&DQRQLFDO&RUUHODWLRQ$QDO\VLV%DFNJURXQG $SSHQGL[%7KH$VVXPSWLRQVRI&&$DQG0DMRU7KUHDWVWR WKH5HOLDELOLW\DQG9DOLGLW\RI5HVXOWV
7KH500(UURU'LVWULEXWLRQ
,QWURGXFWLRQ 'HULYDWLRQRIWKH500(UURU'LVWULEXWLRQ 3URSHUWLHVRIWKH(UURU'LVWULEXWLRQ 9DULDWLRQVRIWKH500(UURU'LVWULEXWLRQ 5HIHUHQFHV
)LWWLQJ3URFHGXUHV IRUWKH(UURU'LVWULEXWLRQ
,QWURGXFWLRQ %ULHI5HYLHZRI&XUUHQW0HWKRGRORJLHV )LWWLQJYLD0RPHQW0DWFKLQJ )LWWLQJYLD4XDQWLOH0DWFKLQJ 7ZR1XPHULFDO([DPSOHV $PRPHQWPDWFKLQJH[DPSOH $TXDQWLOHPDWFKLQJH[DPSOH 5HIHUHQFHV
(VWLPDWLQJWKH(UURU'LVWULEXWLRQ
,QWURGXFWLRQ 3HUFHQWLOH%DVHG(VWLPDWLRQ 7KHHVWLPDWLRQSURFHGXUH 7ZRQXPHULFDOH[DPSOHV SHUFHQWLOHEDVHGHVWLPDWLRQ 0RPHQW%DVHG(VWLPDWLRQ ,QWURGXFWLRQ 3URFHGXUH, 3URFHGXUH,, 7ZRQXPHULFDOH[DPSOHV PRPHQWEDVHGHVWLPDWLRQ 5HIHUHQFHV
&217(176
6SHFLDO&DVHVRIWKH5000RGHO
[YLL
&XUUHQW5HODWLRQDO0RGHOVDV6SHFLDO&DVHVRI500 &KHPLVWU\DQG&KHPLFDO(QJLQHHULQJ 3K\VLFV (OHFWULFDOHQJLQHHULQJ *URZWKPRGHOV &XUUHQW0RGHOVRI5DQGRP9DULDWLRQDV5000RGHOV 7KH-RKQVRQIDPLOLHVRIGLVWULEXWLRQV 7XNH\JDQGK6\VWHPVRIGLVWULEXWLRQV )LVKHU VWUDQVIRUPDWLRQRIWKHVDPSOHFRUUHODWLRQ +DOGDQHSRZHUWUDQVIRUPDWLRQDQG:LOVRQ+LOIHUW\ DSSUR[LPDWLRQWRχ2 %R[&R[QRUPDOL]LQJWUDQVIRUPDWLRQ &DXFK\GLVWULEXWLRQ *HQHUDOL]HG,QYHUVH*DXVVLDQGLVWULEXWLRQDQGWKH/HY\ GLVWULEXWLRQ *HQHUDOL]HG*DPPDGLVWULEXWLRQV 5HIHUHQFHV
(YDOXDWLQJ500IRU&RPSOLDQFH
,QWURGXFWLRQ &RPSOLDQFHIRU0RGHOLQJ6\VWHPDWLF9DULDWLRQ &RPSOLDQFHLQ0RGHOLQJ5DQGRP9DULDWLRQ 5HIHUHQFHV
3$57,,,02'(/,1*6<67(0$7,&9$5,$7,21 $33/,&$7,216 &RPSDUDWLYH6ROXWLRQVIRU5HODWLRQDO0RGHOV
,QWURGXFWLRQ 7ZR1HZ3UREOHPV ([DPSOH7KH:LQGVKLHOG([SHULPHQW ([DPSOH7KH(FRQRPLVW%LJ0DF3DULW\,QGH[ 7ZR)DPLOLDU3UREOHPV &RQW GIURP&KDSWHU ([DPSOH7KH:DYH6ROGHULQJ3URFHVV ([DPSOH7KH5HVLVWLYLW\GDWD
[YLLL
&217(176
&RPSDULVRQRI0RGHOV 0DOORZ V&S $NDLNH V,QIRUPDWLRQ&ULWHULRQ $,& 5HIHUHQFHV
5HOLDELOLW\(QJLQHHULQJ ZLWK&HQVRULQJ
,QWURGXFWLRQ 500(VWLPDWLQJZLWK&HQVRUHG'DWD $1XPHULFDO([DPSOH7KH5)/PRGHO 5HIHUHQFHV
6RIWZDUH5HOLDELOLW\*URZWK0RGHOV
,QWURGXFWLRQ ([DPSOH0XVD V0'DWD6HW ([DPSOH0XVD V0'DWD6HW 5HIHUHQFHV
0RGHOLQJD&KHPR5HVSRQVH
,QWURGXFWLRQ $SSO\LQJ500WRD&KHPR5HVSRQVH)LUVW9DULDWLRQ ([DPSOH7HPSHUDWXUHGHSHQGHQFHRIYDSRU SUHVVXUH ([DPSOH7HPSHUDWXUHGHSHQGHQFHRIVROLG KHDWFDSDFLW\ $SSO\LQJ500WRD&KHPR5HVSRQVH6HFRQG9DULDWLRQ ([DPSOH7HPSHUDWXUHGHSHQGHQFHRIYDSRU SUHVVXUH ([DPSOH+HDWFDSDFLW\RIVROLGVDQGOLTXLGV 2WKHUWHPSHUDWXUHGHSHQGHQWSURSHUWLHV 5HIHUHQFHV
)RUHFDVWLQJ66KDSHG'LIIXVLRQ3URFHVVHV
,QWURGXFWLRQ 7KHRUHWLFDO%DFNJURXQGIRU6VKDSHG'LIIXVLRQ3URFHVVHV 0RGHOLQJDQG)RUHFDVWLQJ6VKDSHG3URFHVVHV
&217(176
[L[
1XPHULFDO([DPSOHV )RUHFDVWLQJ73JLYHQ3 )RUHFDVWLQJ37JLYHQ7 5HIHUHQFHV $SSHQGL[$&XUUHQW)RUHFDVWLQJ0RGHOV $SSHQGL[%'HVFULSWLRQRI'DWD6HWV
3$57,902'(/,1*5$1'209$5,$7,21 $33/,&$7,216 500'LVWULEXWLRQDO$SSUR[LPDWLRQV
,QWURGXFWLRQ )LWWLQJ500ZLWK1RUPDORU/RJQRUPDO(UURUV )LWWLQJZLWKD/RJLVWLF(UURU7HUP $SSUR[LPDWLRQVIRUWKH1RUPDODQGWKH3RLVVRQ 'LVWULEXWLRQV $SSUR[LPDWLQJWKH3RLVVRQTXDQWLOH $SSUR[LPDWLQJWKH&')RIWKH6WDQGDUG1RUPDO 5HIHUHQFHV
,QYHUVH1RUPDOL]LQJ7UDQVIRUPDWLRQV
,QWURGXFWLRQ 'HULYDWLRQRIWKH2ULJLQ,17 )RXU0RPHQW0DWFKLQJ7KH3UREOHPDQGD6ROXWLRQ 3DUDPHWHU5HGXFHG,17V 2IIVSULQJ,17, 2IIVSULQJ,17,, 2IIVSULQJ,17,,, 2IIVSULQJ,17,9 'LVWULEXWLRQ)LWWLQJ3URFHGXUHV )LWWLQJSURFHGXUHVIRU,17, 6HFWLRQ $ILWWLQJSURFHGXUHIRU,17,, 6HFWLRQ $ILWWLQJSURFHGXUHIRU,17,,, 6HFWLRQ )LWWLQJSURFHGXUHVIRU,17,9 6HFWLRQ 1RUPDOL]LQJ7UDQVIRUPDWLRQV 5HIHUHQFHV
[[
&217(176
3LHFH:LVH/LQHDU$SSUR[LPDWLRQV
,QWURGXFWLRQ 7KH%DVLF0RGLILHG 1RUPDO $SSUR[LPDWLRQ $9DULDWLRQRIWKH%DVLF0RGHOZLWKD)LWWLQJ3URFHGXUH $6LPSOLILHG)LWWLQJ3URFHGXUH $)LWWLQJ3URFHGXUH8VLQJ)LUVWDQG6HFRQG'HJUHH 0RPHQWV 5HYLHZRI5HODWHG3XEOLVKHG5HIHUHQFHV $1XPHULFDO([DPSOH 5HIHUHQFHV
*HQHUDO&RQWURO&KDUWV
,QWURGXFWLRQ *HQHUDO&RQWURO6FKHPHVIRU$WWULEXWHV ,QWURGXFWLRQ 0RGLILHGFRQWUROOLPLWVIRUDWWULEXWHV 6LPSOLILHGOLPLWV 3UREDELOLW\OLPLWVZLWKLQIODWHGVNHZQHVV 3UREDELOLW\OLPLWVIRUVRPHDWWULEXWHGLVWULEXWLRQV 1XPHULFDODVVHVVPHQW *HQHUDO&RQWURO6FKHPHVIRU9DULDEOHV ,QWURGXFWLRQ ,17EDVHGFRQWUROVFKHPHVIRUYDULDEOHV 5HIHUHQFHV
,QYHQWRU\$QDO\VLV
,QWURGXFWLRQ )LUVW$SSURDFK7KH4XDQWLOH)XQFWLRQDQGWKH/RVV )XQFWLRQ 6HFRQG$SSURDFK7KH4XDQWLOH)XQFWLRQDQG/RVV )XQFWLRQ )LUVW$SSURDFK1HZVER\3UREOHPZLWK2UGHUXSWR 3ROLF\ 6HFRQG$SSURDFK7ZR([DPSOHV 7KH&RQWLQXRXV5HYLHZ 45 0RGHO 6DIHW\OHDGWLPHVIRUSXUFKDVHGFRPSRQHQWV
&217(176
5HIHUHQFHV 5HYLHZ4XHVWLRQV $XWKRU,QGH[ 6XEMHFW,QGH[
[[L
This page intentionally left blank
&KDSWHU
,QWURGXFWLRQ
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
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5,1752'8&7,21
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³5HVSRQVH´WRLQFOXGHQRW RQO\YDULDEOHVWKDWH[SHULHQFHV\VWHPDWLFYDULDWLRQ QDPHO\UHVSRQGWR YDULDWLRQ LQ UHODWHG IDFWRUV EXW DOVR WR YDULDEOHV WKDW H[SHULHQFH RQO\ UDQGRP YDULDWLRQ QR V\VWHPDWLF YDULDWLRQ LV WUDQVPLWWHG YLD H[WHUQDO DIIHFWLQJ IDFWRUV 7R GHPRQVWUDWH ZKDW LV LPSOLHG E\ WKLV H[WHQVLRQ VXSSRVHWKDWZHLQYHVWLJDWHDSURFHVVYDULDEOHGHILQHGIRUDSURFHVVWKDW LV DVVXPHG WR EH LQ D VWDWH RI VWDWLVWLFDO VWDELOLW\ ³LQ FRQWURO´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³5HVSRQVH´ WR LQFOXGH YDULDEOHV WKDW DUH VXEMHFWHG WR RQO\ UDQGRP YDULDWLRQLVWKXVFRUURERUDWHG &RQVLGHU WKH WHUP LQ WKH VXEWLWOH RI WKH ERRN DOVR JLYHQ LQ WKH RSHQLQJ VHQWHQFH RI WKLV FKDSWHU (PSLULFDO PRGHOLQJ :KDW GR ZH
5(63216(02'(/,1*0(7+2'2/2*<
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¶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
&+$37(5,1752'8&7,21
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³WUXH´ XQGHUO\LQJSURFHVVGLVWULEXWLRQDSUDFWLWLRQHURI63&RU3&$PD\RSWWR HLWKHU HVWDEOLVK WKDW FHUWDLQ FRQGLWLRQV H[LVW WKDW QHFHVVDULO\ JHQHUDWH D VSHFLILHG GLVWULEXWLRQ D JRRG H[DPSOH LV WKH 3RLVVRQ GLVWULEXWLRQ RU VKH PD\ EH LPSOHPHQWLQJ D ³GLVWULEXWLRQ LGHQWLI\LQJ SURFHGXUH´ QRZ DYDLODEOH LQ DQ\ VWDWLVWLFDO DQDO\VLV SDFNDJH LQ RUGHU WR ILQG WKH ³WUXH´ SURFHVV GLVWULEXWLRQ %RWK DSSURDFKHV LQFXU FRQVLGHUDEOH GLIILFXOWLHV ZKLFKZLOOEHDGGUHVVHGODWHULQWKHERRN &KDSWHU $OWHUQDWLYHO\ D SUDFWLWLRQHU PD\ UHVRUW WR HPSLULFDO PRGHOLQJ ,QVWHDGRIDWWHPSWLQJWRILQGWKHH[DFW &RUUHFW GLVWULEXWLRQRQHPD\ XVHDPHPEHURIVRPHNQRZQSDUDPHWHUULFKIDPLO\RIGLVWULEXWLRQVWKDW FDQ EH VKRZQ WR ILW ZHOO WKH DYDLODEOH SURFHVV GDWD *RRGQHVVRIILW FULWHULD PD\ WKHQ MXVWLI\WKHDGRSWLRQ RI WKH VHOHFWHG GLVWULEXWLRQ DV D VDWLVIDFWRU\UHSUHVHQWDWLRQRIWKHXQGHUO\LQJ³XQNQRZQ´GLVWULEXWLRQ1R FODLP LV PDGH WKDW WKLV LV WKH WUXH SURFHVV GLVWULEXWLRQ 1R WKHRU\ LV DVVXPHG WKDW PD\ FRUURERUDWH WKLV ³DGRSWLRQ´ 7KH ZKROH DQDO\VLV LV VROHO\GDWDGULYHQ 6LPLODUO\LQGHYHORSLQJVRIWZDUHUHOLDELOLW\JURZWKPRGHOVZHPD\ DVVXPH IURP WKHRU\ WKDW LQ WHVWLQJ D VRIWZDUH SURGXFW WKH UDWH RI RFFXUUHQFH RU GHWHFWLRQ RI VRIWZDUH HUURUV IROORZV D 3RLVVRQ GLVWULEXWLRQ
5(63216(02'(/,1*0(7+2'2/2*<
GLVWULEXWLRQ PRVW FRPPRQO\ WKH PHDQ RI WKH FXPXODWLYH QXPEHU RI HUURUVGHWHFWHG LVFKDQJLQJDVDIXQFWLRQRIWKHQXPEHURIWHVWLQJKRXUV SXW LQ WKH WHVWLQJ HIIRUW &RQVHTXHQWO\ IRU VRIWZDUH UHOLDELOLW\JURZWK PRGHOVHPSLULFDOPRGHOLQJRIWKHPHDQRIWKH3RLVVRQLQWHUPVRIODERU WHVWLQJ KRXUV LV UHTXLUHG ZKLFK ZLOO EH VKRZQ WR ILW ZHOO DYDLODEOH REVHUYDWLRQV 2XULQDELOLW\WRDOZD\VPRGHODUHVSRQVHE\DYDLODEOHUHOHYDQWWKHRU\ HVSHFLDOO\ LQ WKH HQJLQHHULQJ GLVFLSOLQHV KDV OHG WR WKH GHYHORSPHQW RI JHQHUDO V\VWHPV RI HPSLULFDO PRGHOLQJ $ ZHOONQRZQ JHQHUDO PHWKRGRORJ\ IRU HPSLULFDO PRGHOLQJ RI V\VWHPDWLF YDULDWLRQ QR GRXEW IDPLOLDU WR WKH UHDGHU LV OLQHDU UHJUHVVLRQ DQDO\VLV 6XEMHFW WR VRPH JHQHUDOO\YDOLGVHWRIDVVXPSWLRQVNQRZQWRIRUPWKHQRUPDOVFHQDULR DQH[DFWGHILQLWLRQRIWKLVZLOOEHJLYHQODWHULQWKHERRNLQ&KDSWHU OLQHDUUHJUHVVLRQDOORZVRQHWRPRGHODUHVSRQVH¶VV\VWHPDWLFYDULDWLRQLQ WHUPVRIWKHDIIHFWLQJHIIHFWVVRWKDWJRRGFRPSOLDQFH ILW ZLWKDYDLODEOH REVHUYDWLRQV LV DFKLHYHG
&+$37(5,1752'8&7,21
ZKHUHWKHPRPHQWVRIWKHWUXHXQNQRZQGLVWULEXWLRQDUHHVWLPDWHGIURP DYDLODEOHVDPSOHGDWD 7RGHPRQVWUDWHKRZWKLVSULQFLSOHKDVEHHQDSSOLHGLQWKHSDVWDQG TXLWH VXFFHVVIXOO\ FRQVLGHU WKH ZHOONQRZQ 6KHZKDUW FRQWURO FKDUWV :KHQ 6KHZKDUW XVHG WKH QRUPDO GLVWULEXWLRQ WR EXLOG KLV ³FRQWURO FKDUWV´ EDFN LQ WKH PLGWZHQWLHV RI WKH SUHYLRXV FHQWXU\ ZKDW KH KDG DFWXDOO\ GRQH ZDV WR ILW WKH QRUPDO GLVWULEXWLRQ WR WKH SUHVXPDEO\ ³XQNQRZQ´ GLVWULEXWLRQ 7KLV ZDV GRQH E\ PDWFKLQJ RI WKH ILUVW WZR PRPHQWVRIWKHQRUPDOGLVWULEXWLRQZLWKWKRVHRIWKHWUXH RFFDVLRQDOO\ XQNQRZQ SURFHVV GLVWULEXWLRQ )RU H[DPSOH WKH FRPPRQO\ DSSOLHG S FRQWURO FKDUW XVHG WR PRQLWRU WKH VWDELOLW\ RI D 3HUFHQWDJH YDULDEOH DVVXPHVWKDWWKHSURFHVVGLVWULEXWLRQLVQRUPDOWKRXJKZHNQRZWKDWWKH WUXHXQGHUO\LQJGLVWULEXWLRQLVELQRPLDO(QVXULQJWKDWWKHPHDQDQGWKH YDULDQFHRIWKHQRUPDOGLVWULEXWLRQXVHGWRFRQVWUXFWWKHFRQWUROOLPLWV FRLQFLGHZLWKWKRVHRIWKHELQRPLDOGLVWULEXWLRQRQHFDQUHVWDVVXUHGWKDW LQ WKLV FDVH WKH FRQWURO OLPLWV UHIOHFW ZHOO WKH WUXH GLVWULEXWLRQ RI WKH PRQLWRULQJ VWDWLVWLF (TXDOLW\ ZLWK WKH WUXH SURFHVV GLVWULEXWLRQ WKH ELQRPLDO RI WKH ILUVW WZR PRPHQWV KDV DOORZHG XV WR XVH WKH QRUPDO GLVWULEXWLRQDVDQHPSLULFDOPRGHOIRUWKHELQRPLDOGLVWULEXWLRQ 6KHZKDUW FKDUWV FRQVWLWXWH RQH H[DPSOH RI $ JHQHUDO PHWKRGRORJ\ IRUHPSLULFDOPRGHOLQJWKLVWLPHRIUDQGRPYDULDWLRQZKHUHHTXDOLW\RI PRPHQWV LV WKH XQGHUO\LQJ SULQFLSOH WKDW FRQIHUV OHJLWLPDF\ RQ WKH PHWKRGRORJ\XVHG$VMXGJHGE\WKHJHQHUDODFFHSWDELOLW\DQGGHJUHHRI LPSOHPHQWDWLRQ RI 6KHZKDUW FRQWURO FKDUWV WKLV JHQHUDO HPSLULFDO PRGHOLQJ PHWKRGRORJ\ LV UHDVRQDEO\ HIIHFWLYH LQ FDSWXULQJ WKH XQGHUO\LQJSURFHVVYDULDWLRQ :KHQ WKH SURFHVV GLVWULEXWLRQ LV VR VNHZHG DV\PPHWULF VR WKDW PDWFKLQJ RI RQO\ WKH ILUVW WZR PRPHQWV LV QRW VXIILFLHQW PDWFKLQJ RI PRUHPRPHQWVPD\EHUHTXLUHGIRUH[DPSOHPDWFKLQJRIWKHILUVWIRXU PRPHQWV,QWKLVFDVHDVLQWKHIRUPHUHVWLPDWHVRIWKH³WUXH´PRPHQWV RI UHFDOO WKH XQNQRZQ SURFHVV GLVWULEXWLRQ DUH GHULYHG IURP VDPSOH REVHUYDWLRQVRIWKHPRQLWRUHGSURFHVV &OHPHQWV SURYLGHV DQ H[DPSOH IRU WKH ³PRPHQW PDWFKLQJ DSSURDFK´ ZKHQ WKH XQGHUO\LQJ XQNQRZQ SURFHVV GLVWULEXWLRQ LV DV\PPHWULF )RU WKLV FDVH &OHPHQWV KDV VXJJHVWHG D PHWKRGRORJ\ WR REWDLQSURFHVVFDSDELOLW\LQGLFHVXVLQJDPHPEHURIWKHIRXUSDUDPHWHU
5(63216(02'(/,1*0(7+2'2/2*<
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³(PSLULFDOPRGHOLQJ´PD\EHDQDSSURSULDWHWHUPWRGHILQHWKHFDWHJRU\ WRZKLFKDOOWKHVHPHWKRGRORJLHVEHORQJ :KDWSDUWLFXODUIHDWXUHVVHWHPSLULFDOPRGHOLQJDSDUWIURPVFLHQWLILF DQGVWDWLVWLFDOPRGHOLQJ"&RQVLGHUPRGHOLQJRIV\VWHPDWLFYDULDWLRQILUVW 6FLHQWLILF UHODWLRQDO PRGHOV GHSDUW LQ D VLJQLILFDQW ZD\ IURP FRUUHVSRQGLQJ PRGHOV GHULYHG E\ HPSLULFDO PRGHOLQJ ,Q VFLHQWLILF PRGHOLQJ UDQGRP HUURUV DUH UDUHO\ VSHFLILHG $V HYHQ D VXSHUILFLDO H[DPLQDWLRQ RI WH[WERRNV LQ WKH H[DFW VFLHQFHV ZRXOG UHYHDO +DOOLGD\ HW DO PRVW PRGHOV DGGUHVVHG LQ WKHVH ERRNV DUH DVVXPHG WR EH GHWHUPLQLVWLF ZLWK QR DOORZDQFH IRU UDQGRP HUURUV 7KHUH PLJKW EH D
&+$37(5,1752'8&7,21
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
5(63216(02'(/,1*0(7+2'2/2*<
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¶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
&+$37(5,1752'8&7,21
6HFRQGO\ DQG QRW XQUHODWHG WR WKH ILUVW SRLQW D PDMRU HQGHDYRU LQ VWDWLVWLFV LV WKH FKDUDFWHUL]DWLRQ RI GLVWULEXWLRQV ZKLFK DWWHPSW WR IRUPXODWHFRQGLWLRQVOHDGLQJQHFHVVDULO\WRDFHUWDLQWUXHGLVWULEXWLRQ D ZHOONQRZQ H[DPSOH LV WKH PHPRU\OHVV SURSHUW\ RI WKH H[SRQHQWLDO GLVWULEXWLRQ 0RGHOLQJLQVWDWLVWLFVWKXVLPSOLHVWKDWWKHWUXHGLVWULEXWLRQ LVEHLQJPRGHOHGDQGRQO\DVLQJOHPRGHOPD\DWDFHUWDLQSRLQWLQWLPH EHWKHFRUUHFWRQH,QHPSLULFDOPRGHOLQJRIUDQGRPYDULDWLRQZHGRQRW FDUH DW DOO DERXW WKH WUXH GLVWULEXWLRQ $V ORQJ DV UHVXOWV IURP DSSURSULDWH JRRGQHVVRIILW FULWHULD VXSSRUW WKH DGRSWLRQ RI WKH ILWWHG HPSLULFDO PRGHO ZH DUH KDSS\ ZLWK LW $Q RIWHQFLWHG DSKRULVP DWWULEXWHGWR3URIHVVRU%R[FRPHVWRPLQG $OOPRGHOVDUHZURQJVRPH DUHXVHIXO *LYHQWKHFHQWUDOUROHWKDWHPSLULFDOPRGHOLQJSOD\VLQDOOEUDQFKHVRI HQJLQHHULQJ EXW PRVW QRWDEO\ LQ TXDOLW\ DQG UHOLDELOLW\ HQJLQHHULQJ LW PD\FRPHDVDVXUSULVHWKDWLQGHHGQRVHULRXVHIIRUWVKDYHWKXVIDUEHHQ GHYRWHG WR WKH IRUPXODWLRQ RI UHTXLUHPHQWV WKDW HPSLULFDO PRGHOLQJ VKRXOG IXOILOO ,Q OLJKW RI WKH P\ULDG RI PHWKRGRORJLHV DQG DSSURDFKHV DYDLODEOH WR WKH SUDFWLWLRQHU ZKR ZLVKHV WR HQJDJH LQ HPSLULFDO PRGHOLQJ LW VHHPV KLJKO\ GHVLUDEOH WKDW LQ RUGHU WR EH DEOH WR HYDOXDWH DQG FRPSDUH WKH TXDOLW\ RI DOWHUQDWLYH PHWKRGRORJLHV D VHW RI UHTXLUHPHQWV VKRXOG EH DJUHHG XSRQ DQG FRPSOLDQFH ZLWK WKHVH UHTXLUHPHQWVEHXVHGDVDPHDVXULQJVWLFNWRMXGJHDVVHVVDQGFRPSDUH WKH TXDOLW\ RI YDULRXV DSSURDFKHV WR HPSLULFDO PRGHOLQJ RI V\VWHPDWLF YDULDWLRQDQGRIUDQGRPYDULDWLRQ 7KLV LV GRQH LQ &KDSWHU ZKHUH UHTXLUHPHQWV DUH GHILQHG DQG HYDOXDWLRQ RI FRPSOLDQFH ZLWK WKHVH UHTXLUHPHQWV RI PDMRU PHWKRGRORJLHVIRUHPSLULFDOPRGHOLQJLVFDUULHGRXW ,Q WKH SXUVXLQJ 3DUW , RI WKH ERRN ZH GHVFULEH FXUUHQW PDLQVWUHDP UHODWLRQDO PRGHOV DQG FXUUHQW PDMRU PHWKRGRORJLHV IRU PRGHOLQJ V\VWHPDWLFYDULDWLRQDQGUDQGRP YDULDWLRQ&KDSWHUSUHVHQWVDVDPSOH RI PRGHOV WKDW RQH FDQ ILQG LQ YDULRXV VFLHQWLILF DQG HQJLQHHULQJ GLVFLSOLQHV ZKHUH D PRQRWRQH FRQYH[ UHODWLRQVKLS H[LVWV EHWZHHQ WKH DIIHFWLQJ IDFWRUDQG WKHUHVSRQVH )HDWXUHV FRPPRQ WR DOOWKHVH PRGHOV DUH H[SORUHG LQ &KDSWHU DQG 7KH /DGGHU RI IXQGDPHQWDO PRQRWRQH FRQYH[ IXQFWLRQV LV LQWURGXFHG 7KH ODGGHU ODWHU EHFRPHV RQH RI WKH FRUQHUVWRQHVIRUWKHD[LRPDWLFGHYHORSPHQWRIWKH500PRGHO
5(63216(02'(/,1*0(7+2'2/2*<
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
3DUW,
&855(1702'(/6$1' 02'(/,1*0(7+2'2/*,(6
This page intentionally left blank
&KDSWHU
5HODWLRQDO0RGHOVLQ(QJLQHHULQJDQG WKH6FLHQFHV 0RQRWRQH&RQYH[&RQFDYH 5HODWLRQVKLSV
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
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(55(/$7,21$/02'(/6,1(1*,1((5,1* 6&,(1&(6
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
5(63216(02'(/,1*0(7+2'2/2*<
5LHGHO PRGLILHG E\DGGLQJWZRWHUPVWRREWDLQ ORJ 3 $%7 & ORJ 7 '7(
(TXDWLRQ LVMXGJHGE\'DXEHUW WREHWKHPRVWDGHTXDWH WRGD\IRUPRGHOLQJWHPSHUDWXUHGHSHQGHQFHRIYDSRUSUHVVXUH+RZHYHU WKLV LV EDVLFDOO\ $QWRLQH HTXDWLRQ LPSURYHG HPSLULFDOO\ E\ DGGLQJ WZR SRO\QRPLDOWHUPV )RU VXUIDFH WHQVLRQ 6 'DXEHUW UHFRPPHQGV WKH XVH RI *XJJHQKHLP¶VHPSLULFDOHTXDWLRQ 6 $ 77F & DQG7FLVWKHFULWLFDOWHPSHUDWXUH DFRQVWDQW &RUUHODWLRQ RI KHDW RI YDSRUL]DWLRQ ZLWK WHPSHUDWXUH PXVW EH FRQVLVWHQWZLWKWKHH[DFW&ODSH\URQHTXDWLRQG3G7 ∆+Y 7∆9 ZKHUH 9 LV YROXPH DQG ∆+Y LV WKH KHDW RI YDSRUL]DWLRQ 7KHRUHWLFDO FRQVLGHUDWLRQV UHTXLUH WKDW ∆+Y ZLOO EH PRGHOHG LQ WHUPV RI W 7U ZKHUH7U 77FLVWHPSHUDWXUHUHODWLYHWRLWVYDOXHDWWKHFULWLFDOSRLQW7F 'DXEHU LQWURGXFHVWKUHHFXUUHQWPRGHOV ∆+Y $>%G@ ∆+Y $W%
∆+Y %W$&W $ 'W $ « ZLWKG W& )RU VROLG GHQVLW\ 'DXEHUW QRWHV WKH IROORZLQJ $V WKH YDULDWLRQ ZLWK WHPSHUDWXUH LV QRW VWURQJ D OLQHDU HTXDWLRQ LV QRUPDOO\ DGHTXDWHZLWKDGDWDUDQJHRIWKHORZHVWGDWDSRLQWWRWKHWULSOHSRLQWRI WKH FRPSRXQG ,Q D IHZ FDVHV ZLWK D ODUJH DPRXQW RI GDWD D VLPSOH TXDGUDWLF SRO\QRPLDO ZDV UHFRPPHQGHG )RU PRVW VLWXDWLRQV XVH RI D VLQJOHH[SRQHQWLDOYDOXHRYHUWKHHQWLUHWHPSHUDWXUHUDQJHLVVXIILFLHQW )RU OLTXLG GHQVLW\ ρ/ 5DFNHWW VXJJHVWHG DQ HTXDWLRQ ZKLFK 'DXEHUW FRQFOXGHV LV UHFRPPHQGHG IRU FRUUHODWLRQ RI OLTXLG GHQVLW\5DFNHWW¶VHTXDWLRQLV
ORJ ρ/ $%>7&@'
ZKHUH^$%&'`DUHSDUDPHWHUVWKDWQHHGWREHGHWHUPLQHG
&+$37(55(/$7,21$/02'(/6,1(1*,1((5,1* 6&,(1&(6
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`DUHSDUDPHWHUVZKLFKQHHGWREHGHWHUPLQHG )RU WHPSHUDWXUH GHSHQGHQF\ RI OLTXLG YLVFRVLW\ µ/ 'DXEHUW UHFRPPHQGV
ORJ µ/ $%7&ORJ 7 '7(
7KLVLVDJDLQLQDSRO\QRPLDOIRUP )RU WKHUPDO FRQGXFWLYLW\ 'DXEHUW H[DPLQHV ERWK OLTXLG DQG ORZSUHVVXUH YDSRU WKHUPDO FRQGXFWLYLW\ $OO RI WKH H[DPLQHG UHODWLRQVKLSVDUHHLWKHUSRO\QRPLDOVLQ7RULQW 7U 77F )LQDOO\ WKH IDPRXV $UUKHQLXV HTXDWLRQ GHVFULEHV WKH HIIHFW WKDW WHPSHUDWXUHKDVRQWKHUDWHRIDVLPSOHFKHPLFDOUHDFWLRQ 5 $H[S>(D κ%7 @ ZKHUH 5 LV WKH UHDFWLRQ RU GLIIXVLRQ UDWH 7 LV WHPSHUDWXUH LQ GHJUHHV .HOYLQ (D LV WKH HIIHFWLYH DFWLYDWLRQ HQHUJ\ WKH DPRXQW RI HQHUJ\ UHTXLUHGWRHQVXUHWKDWWKHUHDFWLRQKDSSHQVQRWWHPSHUDWXUHGHSHQGHQW κ%LV%ROW]PDQQ VFRQVWDQWDQG$LVDSURFHVVUHODWHGFRQVWDQW 3K\VLFV 6RPH ZHOONQRZQ UHODWLRQVKLSV GHULYHG IURP 1HZWRQLDQ SULQFLSOHV IURP FXUUHQW FRVPRORJLFDO WKHRULHV RU IURP TXDQWXP PHFKDQLFV DUH
5(63216(02'(/,1*0(7+2'2/2*<
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ρLV
5 ^> π @ 0ρ `
6XSSRVH WKDW WKH WHPSHUDWXUH GLIIHUHQWLDO EHWZHHQ DQ REMHFW DQG LWV VXUURXQGLQJ LV ∆7 7KHQ DFFRUGLQJ WR 1HZWRQ V ODZ RI FRROLQJ LI DW VRPH LQVWDQW W WKH WHPSHUDWXUH GLIIHUHQFH LV ∆7 WKHQ DW VRPH WLPH ODWHU W ! WKH WHPSHUDWXUH GLIIHUHQFH ZLOO EH +DOOLGD\ HW DO S ∆7 ∆7H[S DW (YHU\VROLG DQGOLTXLG DWILQLWHWHPSHUDWXUHLVLQHTXLOLEULXPZLWKLWV RZQYDSRU7KHVDWXUDWLRQYDSRUSUHVVXUH3YDSRUGHSHQGVRQWHPSHUDWXUH 7 PDLQO\ WKURXJK WKH %ROW]PDQQ IDFWRU %HQHGHN DQG 9LOODUV S
3YDSRU∼H[S>(E κ%7 @
ZKHUH κ% LV WKH %ROW]PDQQ IDFWRU 7 LV RQ D .HOYLQ VFDOH DQG (E LV D SDUWLFOHELQGLQJHQHUJ\ ILQGGHWDLOVWKHUHLQ
&+$37(55(/$7,21$/02'(/6,1(1*,1((5,1* 6&,(1&(6
7KH %DURPHWULF IRUPXOD LV D ZHOONQRZQ H[SUHVVLRQ IRU WKH H[SRQHQWLDO GHFUHDVH ZLWK LQFUHDVLQJ KHLJKW RI WKH GHQVLW\ RI D JDV DVVXPLQJD³FRQVWDQW´WHPSHUDWXUH7/HW0EHWKHPROHFXODUZHLJKW5 WKH JDV FRQVWDQW DQG J WKH JUDYLWDWLRQDO FRQVWDQW7KHQ DW KHLJKW ] WKH VSDWLDO GHQVLW\ ρ GHSHQGV RQ WKH UDWLR . ]7 YLD WKH UHODWLRQVKLS %HQHGHNDQG9LOODUVS
ρ . ρ H[S> 0J5 .@
7KH VLPLODULW\ RI DQG WR WKH $UUKHQLXV IRUPXOD ZLWKUHJDUGWRGHSHQGHQFHRQ7LVZRUWKQRWLQJ /HW XV FRQVLGHU VRPH UHODWLRQVKLSV IURP UHODWLYLW\ DQG TXDQWXP SK\VLFV5HVWHQHUJ\SOXVNLQHWLFHQHUJ\IRUDQREMHFWZLWKUHVWPDVVRI 0PRYLQJDWDVSHHGYQHDUWKHVSHHGRIOLJKW&LV IURP(LQVWHLQ V 6SHFLDO7KHRU\RI5HODWLYLW\ ( / 0& / ZKHUH/LVWKHUHODWLYHVSHHGVTXDUHG / Y& 3ODQFN VUDGLDWLRQODZLV +DOOLGD\HWDO S 6 7 πFKλ0 ^H[S> KFλ0N 7 @`
ZKHUH6LVVSHFWUDOUDGLDQF\λ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
3 W 3H[S> NWJHQ W]
5(63216(02'(/,1*0(7+2'2/2*<
(OHFWULFDO(QJLQHHULQJ &RXORPE V ODZ VSHFLILHV WKH HOHFWURVWDWLF IRUFH RI DWWUDFWLRQ RU UHSXOVLRQ )EHWZHHQWZRSRLQWFKDUJHVTDQGTVHSDUDWHGE\GLVWDQFH U,WLVJLYHQE\ +DOOLGD\HWDO ) U N TT U ZKHUHNLVDFRQVWDQW $ FDSDFLWRU ZLWK FDSDFLW\ & ZKLFK LV LQLWLDOO\ XQFKDUJHG T DW W LVFKDUJHGE\DQLGHDOEDWWHU\ZLWKDQHPI HOHFWURPRWLYHIRUFH ( $W WLPH W WKH YROWDJH DFURVV WKH FDSDFLWRU DV PHDVXUHG E\ D UHVLVWRU FRQQHFWHGLQSDUDOOHOZLWKUHVLVWDQFH5LV +DOOLGD\HWDO S
95 W (H[S>W 5& ]
6LPLODUO\DGLVFKDUJLQJFDSDFLWRUZLWKDQLQLWLDO IXOO FKDUJHRIT ZLOOKDYHDWWLPHWDFKDUJHRI T W TH[S>W 5& @ 6LPLODUHTXDWLRQVDSSO\IRUWKHULVH IDOO RIWKHFXUUHQWSURGXFHGE\ DQHPI(ZKLFKLVLQWURGXFHGLQWR UHPRYHGIURP DVLQJOHORRSFLUFXLW FRQWDLQLQJDUHVLVWRU5DQGDQLQGXFWRU/ +DOOLGD\HWDO S 6XSSRVHWKDWHOHFWULFLW\FXUUHQWRILQWHQVLW\ ,UXQVWKURXJKDORRSRI UDGLXV5 DUHD6 7KHPDJQHWLFILHOG% G SURGXFHGDWDSRLQWORFDWHGD GLVWDQFH G DORQJ WKH ORRS D[LV LV SDUDOOHO WR WKH D[LV DQG LV JLYHQ E\ +DOOLGD\HWDO S % G µπ) ,6 G µ) ,5 G ZKHUHµLVDSHUPHDELOLW\FRQVWDQW +DUGZDUH5HOLDELOLW\(QJLQHHULQJ 7KH UHODWLRQVKLS EHWZHHQ IDWLJXH OLIH RI PHWDO FHUDPLF DQG FRPSRVLWH PDWHULDOVDQGDSSOLHGVWUHVVLVDQLPSRUWDQWLQSXWWRGHVLJQIRUUHOLDELOLW\ 0XFK HIIRUW KDV EHHQ GLUHFWHG WRZDUGV PRGHOLQJ WKLV UHODWLRQVKLS $ UHFHQW SDSHU E\ 3DVFXDO DQG 0HHNHU DWWHPSWHG WR PRGHO WKLV UHODWLRQVKLSYLDWKH5DQGRP)DWLJXH/LPLW0RGHO 5)/0 /HW < EH WKHIDWLJXHOLIHDQG V WKH VWUHVV OHYHO7KHPRGHO IRU <LQ WHUPVRIVLV
&+$37(55(/$7,21$/02'(/6,1(1*,1((5,1* 6&,(1&(6
ORJ < β β ORJ Vγ ε
ZKHUHβDQGβDUHIDWLJXHFXUYHFRHIILFLHQWVγLVWKH UDQGRP IDWLJXH OLPLWRIWKHVSHFLPHQDQGεLVDQHUURUWHUP 3DVFXDODQG0HHNHUVXSSRUWHGWKHLUWKHRUHWLFDODQDO\VLVE\DSSO\LQJLW WRIDWLJXHGDWDJLYHQE\6KLPRNDZDDQG+DPDJXFKL 7KH\KDYH DVVXPHGWKDWWKHIDWLJXHOLPLWLVUDQGRPDQGIRUUHDVRQVDVVRFLDWHGZLWK HVWLPDWLRQ RI WKH PRGHO¶V SDUDPHWHUV WKH\ FKDUDFWHUL]H WKH UDQGRP YDULDEOH 9 ORJ γ DIWHU VWDQGDUGL]DWLRQ DV GLVWULEXWHG HLWKHU OLNH WKH VWDQGDUGL]HG VPDOOHVWH[WUHPHYDOXH GLVWULEXWLRQ RU WKH VWDQGDUGL]HG QRUPDOGLVWULEXWLRQ :KHQ V LV FRQVWDQW < H[SHULHQFHV RQO\ UDQGRP IOXFWXDWLRQV DQG WKHVH PD\ EH FDSWXUHG E\ WKH HUURU GLVWULEXWLRQ DVVRFLDWHG ZLWK WKLV PRGHO 3DVFXDO DQG 0HHNHU GHULYH WKLV GLVWULEXWLRQ DQG LQYHVWLJDWH LW EULHIO\7KHDXWKRUVGRQRWVSHFLI\WKHUHODWLRQVKLSEHWZHHQWKHUDQGRP IDWLJXH OLPLW DQG WKH UDQGRP HUURU WHUP ε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µ W ,Q RWKHU ZRUGV WKH SUREDELOLW\ IXQFWLRQ IRU WKH QXPEHURIIDLOXUHV1LLQWKHWLPHLQWHUYDO>WLWL LVJLYHQE\
3 1L Q ^>µ WL µ WL @QQ`H[S^>µ WL µ WL @`
5(63216(02'(/,1*0(7+2'2/2*<
WKDW LV D 3RLVVRQ SUREDELOLW\ IXQFWLRQ ZLWK WLPHGHSHQGHQW SDUDPHWHU µ WL µ WL 1+33PRGHOVGLIIHUE\KRZWKH\PRGHOWKHPHDQIXQFWLRQDQGWKHVH DUHGLYLGHGLQWRWZRPDLQFDWHJRULHV )LQLWHIDLOXUHPRGHOVZKHUHLWLVDVVXPHGWKDWµ ∞ LVILQLWH ,QILQLWHIDLOXUHPRGHOVZKHUHWKHPHDQYDOXHLVQRWERXQGHG ([DPSOHVRIWKHIRUPHUFDWHJRU\DUH , *RHO DQG 2NXPRWR ZKR KDYH DVVXPHG DQ H[SRQHQWLDO PHDQ IXQFWLRQ µ W D>H[S EW @D!E!
,,7KHGHOD\HG6VKDSHGPRGHO EW H[S EW @E!
([DPSOHVRIWKHODWWHUFDWHJRU\DUH ,,, 0XVD HW DO ZKR KDYH SURSRVHG WKH ORJDULWKPLF 3RLVVRQ PRGHO µ W D ORJ EW W!E
,9'XDQH ZKRKDVSURSRVHGWKH3RZHU/DZ3URFHVV 3/3
µ(W)
αWβ , α > , β >
97KHORJSRZHUPRGHOLQWURGXFHGE\;LHDQG=KDR µ W D >ORJ W @ED!E!
9,7KHJHQHUDOL]HGSRZHUIDPLO\PRGHOV *3)0 E\.QDIODQG0RUJDQ µ W α>N W @β
ZKHUH N LV D SRVLWLYH DQG VWULFWO\ LQFUHDVLQJ NQRZQ IXQFWLRQ RI W ZLWK N $UQRX[ HW DO VXJJHVW FULWHULD IRU FKRRVLQJ DQ DSSURSULDWH *3)0PRGHOXVLQJ0XVD VGDWD $ JRRG UHYLHZ RI WKHVH PRGHOV DQG RWKHUV LV JLYHQ LQ ;LH ZKLFKDOVRVHUYHGDVDPDMRUUHIHUHQFHIRUWKHDERYHVKRUWUHYLHZ
&+$37(55(/$7,21$/02'(/6,1(1*,1((5,1* 6&,(1&(6
*URZWK0RGHOV *URZWK PRGHOV DUH D VSHFLDO FDWHJRU\ RI QRQOLQHDU UHJUHVVLRQ PRGHOV 7KH\ DUH XVHG WR GHVFULEH KRZ WKH UHVSRQVH JURZV ZLWK FKDQJHV LQ D UHJUHVVRUYDULDEOH7\SLFDODSSOLFDWLRQVDUHLQELRORJ\ZKHUHSODQWVDQG RUJDQLVPV JURZ ZLWK WLPH EXW WKHUH DUH DOVR PDQ\ DSSOLFDWLRQV LQ HFRQRPLFVDQGHQJLQHHULQJ6RPHVXFKPRGHOVDUHDOVRXVHGLQVRIWZDUH UHOLDELOLW\JURZWKPRGHOLQJ :H SXUVXH KHUH VRPH PDLQVWUHDP PRGHOV DV LQWURGXFHG LQ 0\HUV 0RQWJRPHU\DQG9LQLQJ WKHUHLQ6HFWLRQ 7KHORJLVWLFJURZWKPRGHOLV
\ β1>β2H[S β 3[ @
7KH*RPSHUW]PRGHOLV
\ β1H[S>β2H[S β3[ @
7KH:HLEXOOPRGHOLV
\ β1 β2H[S β3[β4
ZKHUH^β1, β2, β3, β4}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¶HFROH 3RO\WHFKQLTXH >@'DXEHUW7( (YDOXDWHGHTXDWLRQVIRUPVIRUFRUUHODWLQJWKHUPRG\QDPLFVDQG WUDQVSRUW SURSHUWLHV ZLWK WHPSHUDWXUH ,QGXVWULDO DQG (QJLQHHULQJ &KHPLVWU\ 5HVHDUFK
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(55(/$7,21$/02'(/6,1(1*,1((5,1* 6&,(1&(6
>@ ;LH 0 6RIWZDUH UHOLDELOLW\ PRGHOV SDVW SUHVHQW DQG IXWXUH ,Q 5HFHQW $GYDQFHVLQ5HOLDELOLW\7KHRU\0HWKRGRORJ\3UDFWLFHDQG,QIHUHQFH/LPQLRV1 1LNXOLQ0 (GV %LUNKDXVHU%RVWRQ0$
6HFWLRQ >@0\HUV5+0RQWJRPHU\'&9LQLQJ** *HQHUDOL]HG/LQHDU0RGHOV ZLWK$SSOLFDWLRQVLQ(QJLQHHULQJDQGWKH6FLHQFHV-RKQ:LOH\ 6RQV
This page intentionally left blank
&KDSWHU
6KDUHG)HDWXUHVDQG7KH/DGGHU
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
5(63216(02'(/,1*0(7+2'2/2*<
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ε2, DV
ORJ 3 $% 7& ε2.
()
2WKHU IRUPV RI UDQGRP HUURU PD\ EH FRQFHLYHG UHODWLYH WR 3 DVVXPLQJ D FRQVWDQW 7 IRU H[DPSOH ORJ 3ε2 UHIHU WR &KDSWHU IRU GHWDLOV 1RZ GLVFDUG WKH DVVXPSWLRQ WKDW 7 LV FRQVWDQW DQG DVVXPH WKDW 7 LWVHOIIOXFWXDWHVUDQGRPO\,WLVQDWXUDOWRDVVXPHWKDWUDQGRPGHYLDWLRQV LQWHPSHUDWXUHZLOOEHWUDQVPLWWHGWRWKHPHDVXUHGSUHVVXUHRULQRWKHU ZRUGVWKDWDVHFRQGVRXUFHRIUDQGRPYDULDWLRQDIIHFWVWKHUHVSRQVH 'HQRWH WKH UDQGRP GHYLDWLRQ DVVRFLDWHG ZLWK WHPSHUDWXUH E\ ε1. 7HPSHUDWXUH PD\ WKHQ EH UHH[SUHVVHG DV 7 µ7 ε1 ZKHUH µ7 LV WKH H[SHFWHG YDOXH PHDQ RI WKH PHDVXUHG 7 $QWRLQH HTXDWLRQ ZLWK WKH HUURUVDGGHGEHFRPHV
ORJ 3 $% µ7ε& ε2.
(
&+$37(56+$5(')($785(6$1'7+(/$''(5
:HUHODWHWRYDULDWLRQDVVRFLDWHGZLWK7 HLWKHUV\VWHPDWLFFDXVHGE\ FKDQJHV LQ µ7 RU UDQGRP UHSUHVHQWHG E\ ε DV ([WHUQDOO\JHQHUDWHG YDULDWLRQ /LNHZLVH ZH UHODWH WR YDULDWLRQ DVVRFLDWHG ZLWK ε2 DV 6HOI JHQHUDWHGYDULDWLRQ.%RWKRIWKHVHWHUPVDUHVHOIH[SODQDWRU\JLYHQWKH H[SODQDWLRQVH[SRXQGHGHDUOLHU 1H[WFRQVLGHUWKHVSHFLDOVWUXFWXUHRIWKLVPRGHO2EVHUYLQJ LW LV REYLRXV WKDW WKH UHODWLRQVKLS EHWZHHQ WKH UHVSRQVH 3 DQG WKH DIIHFWLQJIDFWRU 7 LVPRQRWRQH DOZD\VLQFUHDVLQJ DQGWKDWLWPD\EH FKDUDFWHUL]HGDV([SRQHQWLDO3RZHU ([DPSOH &KHPLFDO(QJLQHHULQJ ,Q(T *XJJHQKHLP¶VHPSLULFDOHTXDWLRQIRUVXUIDFHWHQVLRQ6ZDV JLYHQ :ULWWHQ KHUH ZLWK WKH HUURUV DV WKH ODWWHU ZHUH H[SODLQHG LQ ([DPSOH LWEHFRPHV
6 $> µΤ + ε1)7F@& ε2)
ZKHUH7FLVWKHFULWLFDOWHPSHUDWXUH DFRQVWDQW 1RWHWKDWWKLVWLPHWKH HUURU ε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
5 7 5DH[S^%> µ7ε1 7D@ε2`7 µ7ε1!7D
5(63216(02'(/,1*0(7+2'2/2*<
)URP WKHUHODWLRQVKLSEHWZHHQWKHUHVSRQVH5DQG7PD\EH FKDUDFWHUL]HGDV([SRQHQWLDO3RZHUUHODWLRQVKLS $QDO\]LQJWKHDERYHWKUHHH[DPSOHV VLPLODUDQDO\VHVPD\LQIDFWEH DSSOLHG WR DQ\ VXEVHW RI WKH PRGHOV LQ &KDSWHU WZR REVHUYDWLRQV HPHUJHZKLFKRQFHFDSWXUHGLQDJHQHUDOPRGHOPD\SURYLGHWKHEDVLV IRUWKHGHVLUHGHPSLULFDOPRGHOLQJPHWKRGRORJ\GHYHORSHGLQ&KDSWHU 2%6(59$7,21$ 9DULDWLRQLQWKHUHVSRQVHPD\EHDWWULEXWHGWRWZRVRXUFHVRIYDULDWLRQ
6HOIJHQHUDWHGYDULDWLRQ 7KLV FRQVLVWV RI UDQGRP YDULDWLRQLQ WKH UHVSRQVH HPDQDWLQJ IURP LWV QDWXUDO LQVWDELOLW\ RU IURP PHDVXUHPHQW HUURU DVVRFLDWHG ZLWK PHDVXULQJ WKH UHVSRQVH 7KLV FRPSRQHQW RI YDULDWLRQLVUHSUHVHQWHGLQWKHDERYHH[DPSOHVE\WKHHUURUWHUPε2.
([WHUQDOO\JHQHUDWHGYDULDWLRQ7KLVFRPSULVHVWZRFRPSRQHQWV 6\VWHPDWLF YDULDWLRQ ZKLFK UHIOHFWV YDULDWLRQ WUDQVPLWWHG WR WKH UHVSRQVH YLD YDULDWLRQ LQ WKH PHDQ RI WKH DIIHFWLQJ IDFWRU V\VWHPDWLF FKDQJHVRIWKHWHPSHUDWXUHPHDQµ7LQWKHDERYHH[DPSOHV ,IVHYHUDO IDFWRUVLQIOXHQFHWKHUHVSRQVHLWLVDVVXPHGWKDWV\VWHPDWLFYDULDWLRQLQ WKH/LQHDU3UHGLFWRU³ /3DOLQHDUFRPELQDWLRQRIWKHDIIHFWLQJHIIHFWV LVJHQHUDWLQJV\VWHPDWLFYDULDWLRQLQWKHUHVSRQVH 5DQGRP YDULDWLRQ ZKLFK UHIOHFWV YDULDWLRQ WUDQVPLWWHG WR WKH UHVSRQVHYLDUDQGRPGHYLDWLRQVLQWKHDIIHFWLQJIDFWRU UDQGRPFKDQJHV LQWHPSHUDWXUHLQWKHDERYHH[DPSOHV RUUDQGRPFKDQJHVLQWKH/3IRU WKHPXOWLSOHIDFWRUFDVH7KLVYDULDWLRQLVUHSUHVHQWHGLQWKHPRGHOVE\DQ DGGLWLYHUDQGRPHUURUε1 2%6(59$7,21% &HUWDLQ PRGHO VWUXFWXUHV RU SDWWHUQV ZKLFK DUH VXSSRVHG WR FDSWXUH XQLIRUPO\ FRQYH[ UHODWLRQVKLSV UHSHDWHGO\ DSSHDU LQ PRGHOV GHYHORSHG LQGHSHQGHQWO\ LQ D P\ULDG RI VFLHQWLILF DQG HQJLQHHULQJ GLVFLSOLQHV 7ZR H[DPSOHV IRU VXFK SDWWHUQV ZHUH SURYLGHG HDUOLHU 3RZHU DQG
&+$37(56+$5(')($785(6$1'7+(/$''(5
([SRQHQWLDO3RZHU UHODWLRQVKLSV 2WKHU UHODWLRQVKLSV OLNH OLQHDU H[SRQHQWLDO RU H[SRQHQWLDOH[SRQHQWLDO DERXQG LQ WKH HQJLQHHULQJ DQG VFLHQWLILF OLWHUDWXUH DV WKH VXUYH\ RI &KDSWHU FOHDUO\ DWWHVWV ,Q IDFW WKHVHYDULRXVEDVLFIXQFWLRQVPD\EHDUUDQJHGLQDKLHUDUFKLFDORUGHUWKDW UHIOHFWVDJUDGXDOO\LQFUHDVLQJ³LQWHQVLW\´RIFRQYH[LW\ 7KLVKLHUDUFK\LVFDSWXUHGLQWKH³/DGGHURI)XQGDPHQWDO8QLIRUPO\ &RQYH[)XQFWLRQV´7KH/DGGHUZLOOEHH[SRXQGHGLQGHWDLOLQWKHQH[W 6HFWLRQ2QFHDPRGHOLVFRQVWUXFWHGWKDWLQFOXGHVDVVSHFLDOFDVHVDOO WKH ³VWHSV³ RI WKH ³/DGGHU³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εLVRIWHQSURSRUWLRQDOWRWKH RUGHU RI PDJQLWXGH RI WKH UHVSRQVH LWVHOI 6XFK LV RIWHQ WKH FDVH LQ PRGHOLQJ D FKHPRUHVSRQVH DV HYLGHQW IURP WKH PRGHOV LQ &KDSWHUV DQG7KHJHQHUDOPRGHOWREHGHYHORSHGLQ&KDSWHUVKRXOGDOORZ
5(63216(02'(/,1*0(7+2'2/2*<
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³/DGGHU´RI6HFWLRQ 7KHDERYHWZRJHQHUDOREVHUYDWLRQVZRXOGIRUPDODUJHSDUWRIWKH IRXQGDWLRQVRQZKLFKWKH500PRGHOLVGHYHORSHGLQ&KDSWHU 7KH/DGGHURI)XQGDPHQWDO8QLIRUPO\&RQYH[&RQFDYH )XQFWLRQV 7KH PRGHOV RI &KDSWHU GHPRQVWUDWH WKDW WKHUH DUH LQGHHG VRPH IXQGDPHQWDO FRQYH[ IXQFWLRQV WKDW UHSHDWHGO\ DSSHDU LQ PRGHOV ZLWK PRQRWRQH FRQYH[FRQFDYH UHODWLRQVKLSV 7KHVH IXQFWLRQV PD\ EH DUUDQJHGLQDKLHUDUFKLFDOIDVKLRQZKLFKH[SUHVVHVDJURZLQJ³LQWHQVLW\ RI FRQYH[LW\³ DV RQH PRYHV IURP RQH ³VWHS³ LQ WKLV KLHUDUFK\ WR WKH QH[W 7KXV DQ H[SRQHQWLDO IXQFWLRQ LV REYLRXVO\ PRUH FRQYH[ WKDQ D SRZHUIXQFWLRQDQGDQH[SRQHQWLDOSRZHUIXQFWLRQLVPRUHFRQYH[WKDQ MXVWDQH[SRQHQWLDOIXQFWLRQ DFRPSDULVRQRIWKHUHODWLYHPDJQLWXGHRI WKH VHFRQG GHULYDWLYH DSSURSULDWHO\ VWDQGDUGL]HG WR DFFRXQW IRU VFDOH PD\SURYLGHDJRRGPHDVXUHIRUWKLVLQWHQVLW\RIFRQYH[LW\
&+$37(56+$5(')($785(6$1'7+(/$''(5
:H GHQRWH WKH KLHUDUFK\ RI PRQRWRQH FRQYH[FRQFDYH UHODWLRQVKLSV 7KH/DGGHURI)XQGDPHQWDO8QLIRUPO\&RQYH[&RQFDYH)XQFWLRQV 7KH /DGGHU LV SUHVHQWHG EHORZ ZKHUH 9 VWDQGV IRU WKH OLQHDU SUHGLFWRU η) SOXVWKHDVVRFLDWHGUDQGRPHUURUε1$OVRIRUHDFKIXQFWLRQ WKH VHFRQG GHULYDWLYH LV JLYHQ WR DOORZ WKH UHDGHU DVVHVV KRZ FHUWDLQ SDUDPHWHUV GHWHUPLQH ZKHWKHU WKH H[SUHVVLRQ LV XQLIRUPO\ FRQYH[ RU FRQFDYH
/LQHDULQFUHDVH7KLVLVDFRQYH[UHODWLRQVKLSZLWKWKHVPDOOHVWUDWH RILQFUHDVHIRUWKHVORSHUHSUHVHQWLQJDVHFRQGGHULYDWLYHRI]HUR
9 ηε1
3RZHULQFUHDVH7KLVUHSUHVHQWVDPRQRWRQHLQFUHDVHRIWKHIRUP9N ZKHUHNLVDUHDOYDOXHGSDUDPHWHU N≠ 7KHVHFRQGGHULYDWLYHLV N N 9 N ([SRQHQWLDO LQFUHDVH 7KLV UHSUHVHQWV D PRQRWRQH LQFUHDVH RI WKH IRUPH[S 9 7KHVHFRQGGHULYDWLYHLV H[S 9 ([SRQHQWLDO3RZHU LQFUHDVH 7KLV UHSUHVHQWV D PRQRWRQH LQFUHDVH RIWKHIRUPH[S 9N 7KHVHFRQGGHULYDWLYHLV N9 N H[S 9N ^ N N9N` ([SRQHQWLDO([SRQHQWLDO LQFUHDVH 7KLV UHSUHVHQWV D PRQRWRQH LQFUHDVHRIWKHIRUPH[S>DH[S 9 @ZKHUH³D´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
5(63216(02'(/,1*0(7+2'2/2*<
PHWKRGRORJ\ IRU PRGHOLQJ XQLIRUPO\ FRQYH[FRQFDYH UHODWLRQVKLSV WR KDYH WKH GHVLUDEOH SURSHUW\ RI EHLQJ DEOH WR DFFRPPRGDWH DW OHDVW WKH IXQFWLRQVLQFOXGHGLQWKH/DGGHU´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
&KDSWHU
$SSURDFKHVWR0RGHO6\VWHPDWLF9DULDWLRQ
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
5(63216(02'(/,1*0(7+2'2/2*<
7R GHPRQVWUDWH WKHVH WZR FRPSRQHQWV FRQVLGHU WZR RI WKH DSSURDFKHV VXUYH\HG LQ WKLV FKDSWHU /LQHDU UHJUHVVLRQ DQDO\VLV DQG WKH %R[&R[ %& SRZHUWUDQVIRUPDWLRQ %R[DQG&R[ ,QOLQHDUUHJUHVVLRQWKHUHODWLRQVKLSEHWZHHQWKH/3DQGWKHUHVSRQVH LV DVVXPHG DSULRUL ,W LV D OLQHDU UHODWLRQVKLS 7KHUHIRUH ZKDW LV UHTXLUHGRIWKHPRGHOLQJSURFHVVLVWRGHWHUPLQHWKHVWUXFWXUHRIWKH/3 ZKLFKHIIHFWVDUHLQFOXGHG DQGHVWLPDWHLWVSDUDPHWHUV &RQVLGHUWKH%&WUDQVIRUPDWLRQ ZLWKSDUDPHWHUλ≠ DSSOLHGWRWKH UHVSRQVH<
<λ λ η ε
ZKHUHε LVDUDQGRP QRUPDO DGGLWLYHHUURUη LVWKH/3
η ββ;β;βN;N
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λ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
&+$37(5$3352$&+(67202'(/6<67(0$7,&9$5,$7,21
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
\ β0 + β1;β11;β12;;
FDQ EH DQDO\]HG YLD OLQHDU UHJUHVVLRQ DQDO\VLV SURYLGHG WKH QRUPDO
5(63216(02'(/,1*0(7+2'2/2*<
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
<
N
N
M
M
M ; M MM ; M
N
L M
LM
; L ; M
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
&+$37(5$3352$&+(67202'(/6<67(0$7,&9$5,$7,21
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
5(63216(02'(/,1*0(7+2'2/2*< < < λ ORJ <
6LQFHWKHVFDOHRI< λ FKDQJHVZLWKλ, QRFRPSDULVRQRIUHVLGXDOVXP RIVTXDUHV566 λ LVSRVVLEOHZKHQZHZLVKWRFRPSDUHVHYHUDOYDOXHV IRU WKH SDUDPHWHU λ PLQLPL]DWLRQ RI 566 LV GHVLUHG 7KHUHIRUH LW LV FXVWRPDU\ WR VWDQGDUGL]H WKH WUDQVIRUPDWLRQ E\ GLYLGLQJ < λ E\ >JP < @λ−1, ZKHUH JP < LV WKH JHRPHWULF PHDQ RI WKH Q REVHUYDWLRQV QDPHO\ JP < <<«JP < @λ−1QRZDOORZVFRPSDULVRQRI 566 REWDLQHG ZLWK YDULRXV YDOXHV RI λ ILQG MXVWLILFDWLRQ IRU WKLV IRU H[DPSOHLQ'UDSHUDQG6PLWK 6HFWLRQ :KLOH WKH %& SURFHGXUH DLPV DW QRUPDOL]LQJ WKH HUURUV YLD D 0/ SURFHGXUH ZKLFKLVHTXLYDOHQWWRPLQLPL]LQJWKH566 RQHPD\ZLVKWR WUDQVIRUP WKH UHVSRQVH LQ RUGHU WR VWDELOL]H WKH YDULDQFH 2U HOVH RQH PD\ZLVKWRDWWDLQOLQHDULW\$V&RRNDQG:HLVEHUJ FRPPHQW ,Q D VXUSULVLQJO\ ODUJH QXPEHU RI UHJUHVVLRQV OLQHDULW\ DQG YDULDQFH VWDELOL]DWLRQ FDQ EH DFKLHYHG ZLWK WKH VDPH WUDQVIRUPDWLRQ WKHUHLQS &RPPHQW$QLQWHUHVWLQJUHODWLRQVKLSEHWZHHQWKH%&SDUDPHWHUλ,DQG WKH UHVSRQVH WUDQVIRUPDWLRQ QHHGHG IRU VWDELOL]LQJ WKH YDULDQFH FDQ EH HVWDEOLVKHG6XSSRVHWKDWWKHUHODWLRQVKLSEHWZHHQWKHPHDQµ<DQGWKH YDULDQFHσ<LVRIWKHW\SH
σ< α (µ<)β
/HW7 < EHDUHVSRQVHWUDQVIRUPDWLRQZKLFKVWDELOL]HVWKHYDULDQFH 'HYHORSLQJ7 < LQWRD7D\ORUVHULHVDURXQGWKHUHVSRQVHPHDQµ<ZH REWDLQ
7 < 7 µ< 7 (µ<) <µ<)«
ZKHUH GLIIHUHQWLDWLRQ RI 7 LV ZLWK UHVSHFW WR < $SSO\LQJ D YDULDQFH RSHUDWRUWRERWKVLGHVZHREWDLQLQWURGXFLQJIRUσ<IURP
7 µ< σ7σ< σ7 [α(µ<)β]
&+$37(5$3352$&+(67202'(/6<67(0$7,&9$5,$7,21
ZKHUHσ7LVWKHYDULDQFHRIWKHWUDQVIRUPHG<7 < 6LQFHE\GHILQLWLRQ WKH YDULDQFH RI WKH GHVLUDEOH 7 < GRHV QRW GHSHQG RQ WKH PHDQ WKH WUDQVIRUPDWLRQ LV VXSSRVHG WR VWDELOL]H WKH YDULDQFH ZH REWDLQ E\ LQWHJUDWLQJRI ZLWKUHVSHFWWRWKHPHDQ
7 µ< .µ<−β+1
7 < .<−β+1
RU
ZKHUH.LVDSDUDPHWHU,WLVHDV\WRUHDOL]HWKHUHODWLRQVKLSEHWZHHQWKH %R[&R[WUDQVIRUPDWLRQSDUDPHWHUλDQGβ,I<λ LVWKHUHTXLUHG%R[ &R[WUDQVIRUPDWLRQWKHQWRHQVXUHWKDWWKLVWUDQVIRUPDWLRQDOVRVWDELOL]HV WKHYDULDQFHZHQHHGWRKDYH
λ
1− β
$ VLPSOH PHWKRG WR LGHQWLI\ WKH UHTXLUHGβ PD\ EH GHYHORSHGIURP 7DNHORJRIERWKVLGHVWRREWDLQ ORJ σ< α0 + β ORJ µ< ZKHUHα0= ORJ α %\GUDZLQJPVDPSOHVIURPWKHVDPHSRSXODWLRQDQG FDOFXODWLQJ WKH PHDQV DQG YDULDQFHV IRU WKHVH VDPSOHV ZH REWDLQ P REVHUYDWLRQV ^µL σL` L P ZKHUH µL DQG σL DUH WKH LWK VDPSOH HVWLPDWHV RI WKH PHDQ DQG WKH VWDQGDUG GHYLDWLRQ UHVSHFWLYHO\ :LWK OLQHDU UHJUHVVLRQ DQ HVWLPDWH RI β PD\ QRZ EH IRXQG IURP WKDW VWDELOL]HV WKH YDULDQFH ,I WKLV HVWLPDWH DSSUR[LPDWHO\ PDLQWDLQV ZKHUH λ ZDV HVWLPDWHG E\ WKH FRQYHQWLRQDO %R[&R[ PD[LPXP OLNHOLKRRG SURFHGXUH WKHQ ZH FDQ EH DVVXUHG WR D ODUJH H[WHQW WKDW WKH %&WUDQVIRUPDWLRQDOVRVWDELOL]HVWKHYDULDQFH +RZZHOOGRHVWKH%&WUDQVIRUPDWLRQIDUHDVDJHQHUDOSODWIRUPIRU HPSLULFDO PRGHOLQJ RI V\VWHPDWLF YDULDWLRQ" $VVXPLQJ WKDW KDV DWWDLQHGWKHQRUPDOVFHQDULRZHPD\PRGHOWKHWUDQVIRUPHGUHVSRQVHE\ < λ ηε ZKHUH η LV WKH /3 DQG ε LV DQ HUURU WHUP QRUPDOO\ DQG LQGHSHQGHQWO\ GLVWULEXWHGZLWKFRQVWDQWYDULDQFH 0RGHOLQJYLD LVUHODWLYHO\VWUDLJKWIRUZDUGLIWKHVWUXFWXUHRIWKH OLQHDU SUHGLFWRU LV GHWHUPLQHG LQ DGYDQFH QDPHO\ WKH SUHGLFWRUV WR EH LQFOXGHGDUHNQRZQ,QWKDWFDVHDYDOXHRIλLVGHWHUPLQHGWKHVDPSOH
5(63216(02'(/,1*0(7+2'2/2*<
UHVSRQVH REVHUYDWLRQV DUH WUDQVIRUPHG OLQHDU UHJUHVVLRQ LV DSSOLHG WR HVWLPDWH WKH /3 DQG WKH DVVRFLDWHG 566 FDOFXODWHG 7KH RSWLPDO SDUDPHWHUλLVWKHQGHWHUPLQHGZKLFKPLQLPL]HVWKH566%R[DQG&R[ KDYH VKRZQ KRZ WKH WUDQVIRUPDWLRQ SDUDPHWHU λ PD\ EH HVWLPDWHG FRQFXUUHQWO\ ZLWK WKH RWKHU PRGHO SDUDPHWHUV RYHUDOO PHDQ DQG WUHDWPHQW HIIHFWV +RZHYHU WKH VWUXFWXUH RI WKH /3 LV VSHFLILHG LQ DGYDQFH ,WLVDOWRJHWKHUDGLIIHUHQWVFHQDULRLIVWHSZLVHUHJUHVVLRQLVGHVLUDEOH ,QWKDWFDVHDUHUXQRIVWHSZLVHOLQHDUUHJUHVVLRQRQWUDQVIRUPHGYDOXHV PD\ UHVXOW LQ D GLIIHUHQW /3 WKDW ZRXOG UHQGHU WKH VHOHFWLRQ RI λ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
&+$37(5$3352$&+(67202'(/6<67(0$7,&9$5,$7,21
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
I \ θ, φ) = H[S^>\θ − E θ @D φ F \φ `
ZKHUHD E DQGF DUHVSHFLILFIXQFWLRQV 7KHSDUDPHWHUθLVDQDWXUDOORFDWLRQSDUDPHWHUDQGφLVRIWHQWHUPHG DGLVSHUVLRQSDUDPHWHU7KHIXQFWLRQD φ LVJHQHUDOO\RIWKHIRUPD φ φω ZKHUH ω LV D NQRZQ FRQVWDQW )RU VRPH FRPPRQ PHPEHUV RI WKH
5(63216(02'(/,1*0(7+2'2/2*<
H[SRQHQWLDO IDPLO\ OLNH WKH ELQRPLDO DQG 3RLVVRQ φ +RZHYHU VLWXDWLRQV RFFXU LQ ZKLFK WKH DFWXDO UHVSRQVH GLVSHUVLRQ LV ODUJHU WKDQ WKDWH[SHFWHGE\WKHDVVXPHGHUURUGLVWULEXWLRQ7KLVUHVXOWVLQDVLWXDWLRQ RIRYHUGLVSHUVLRQ)RUH[DPSOHWKHELQRPLDOSDUDPHWHU3PD\YDU\VR WKDW WKH DFWXDO YDULDQFH LV ODUJHU WKDQ WKDW SUHGLFWHG E\ WKH ELQRPLDO PRGHO,QWKHVHFDVHVZHKDYHφ! 7KHGLVSHUVLRQSDUDPHWHUφLVDQDGGLWLRQDOSDUDPHWHUWKDWWKHDQDO\VW ZLOORFFDVLRQDOO\KDYHWRHVWLPDWHIURPWKHGDWD 6RPHGLVWULEXWLRQVPRVWFRPPRQO\HQFRXQWHUHGLQSUDFWLFHEHORQJWR WKH H[SRQHQWLDO IDPLO\ ([DPSOHV DUH WKH ELQRPLDO WKH QHJDWLYH ELQRPLDO WKH 3RLVVRQWKHQRUPDO LQYHUVH *DXVVLDQJDPPD LQFOXGLQJ WKHH[SRQHQWLDO DQG3DUHWR+RZHYHUVRPHRWKHUGLVWULEXWLRQVLQZLGH XVHOLNH:HLEXOODUHQRWPHPEHUVRIWKHH[SRQHQWLDOIDPLO\ /HW XV GHPRQVWUDWH IRU VRPH RI WKHVH GLVWULEXWLRQV )RU WKH QRUPDOGLVWULEXWLRQ
I \ µ, σ πσ2)−1/2 H[S> \µ σ2)] =
H[S^ \µ − µ2/)/σ2 − (1 >\σ2 + ORJ πσ2)]}.
,W LV FOHDU WKDW WKLV GHQVLW\ LV RI WKH IRUP ZLWK θ µ E θ µ2/2, D φ φφ= σ2DQG
F \φ (1 >\σ2 + ORJ πσ2)]
2QHPD\UHDOL]HZK\θZDVWHUPHGWKHQDWXUDOORFDWLRQSDUDPHWHUDQG φLVDGLVSHUVLRQ RUVFDOH SDUDPHWHU 6LPLODUO\IRUWKH3RLVVRQGLVWULEXWLRQ
I \ µ, σ Hµµ\\
H[S>\ORJ µ µORJ \ @
)RUWKLVGLVWULEXWLRQ
θ ORJ µ)E θ Hθ µDQGF \φ ORJ \
7KHQDWXUDOORFDWLRQSDUDPHWHULVORJ µ) DQGWKHVFDOHSDUDPHWHULVφ :H ZLOO QRW GHOYH KHUH LQWR WKH GHWDLOV RI KRZ HVWLPDWHV RI WKH SDUDPHWHUVRIWKH*/0PRGHODUHGHULYHG+RZHYHULWLVLPSRUWDQWIRU WKH UHDGHU WR XQGHUVWDQG KRZ D WUDQVLWLRQ IURP D PRGHO RI UDQGRP YDULDWLRQ QDPHO\ D PRGHO IRU WKH UHVSRQVH GLVWULEXWLRQ WR D PRGHO RI V\VWHPDWLF YDULDWLRQ LV LPSOHPHQWHG E\ */0 7KLV WUDQVLWLRQ LV QRW
&+$37(5$3352$&+(67202'(/6<67(0$7,&9$5,$7,21
XQOLNHWKHRQHFDUULHGRXWLQOLQHDUUHJUHVVLRQDQDO\VLV,QWKHODWWHULWLV DVVXPHGWKDWWKHUHVSRQVHKDVDQRUPDOGLVWULEXWLRQ
θ ORJ µ J µ η,
µ J η Hη
RU
7KH IXQFWLRQ J µ LV GHQRWHG WKH OLQN IXQFWLRQ VLQFH LW SURYLGHV WKH OLQNEHWZHHQWKHUHVSRQVHPHDQDQGWKH/3$VJLYHQLQ QDPHO\ ZKHQθ J µ ηJ µ LVWHUPHGWKHFDQRQLFDOOLQN ,WKDVVRPHJRRG WKHRUHWLFDO SURSHUWLHV WKDW ZLOO QRW EH GLVFXVVHG KHUH 1RWH WKDW XQOLNH WUDQVIRUPDWLRQDO DSSURDFKHV ZKHUH WKH UHVSRQVH LV FRPPRQO\ DVVXPHG WREHWUDQVIRUPHGLQ*/0LWLVWKHUHVSRQVHPHDQWKDWLVWUDQVIRUPHGLQ RUGHUWRREWDLQWKH/3 $SUDFWLWLRQHUQHHGQRWQHFHVVDULO\XVHDFDQRQLFDOOLQNIXQFWLRQ)RU H[DPSOH WKH %R[&R[ WUDQVIRUPDWLRQ PD\ EH XVHG WR PRGHO WKH OLQN EHWZHHQWKHPHDQDQGWKH/3
η ORJ
7KLV IRUP RI WKH OLQN IXQFWLRQ UHTXLUHV VRPH SUHOLPLQDU\ H[SORUDWLYH LQYHVWLJDWLRQWRGHWHUPLQHWKHEHVWOLQN WKHEHVWSDUDPHWHUλ 2QFHWKHOLQNIXQFWLRQLVGHWHUPLQHGWKHPRGHOHULVUHTXLUHGWRPDNH DQRWKHUGHFLVLRQIUHTXHQWO\LQGHSHQGHQWO\RIWKHVHOHFWHGOLQNIXQFWLRQ DQG WKDW LV WKH GLVWULEXWLRQ RIWKHUHVSRQVH 2EYLRXVO\ LIWKHFDQRQLFDO OLQNLVVHOHFWHGWKHVHGHFLVLRQVDUHQRWFDUULHGRXWLQGHSHQGHQWO\RIHDFK
5(63216(02'(/,1*0(7+2'2/2*<
RWKHU 7KLV GHFLVLRQ GHWHUPLQHV WKH UHODWLRQVKLS EHWZHHQ WKH UHVSRQVH YDULDQFHDQGWKHUHVSRQVHPHDQ5HPHPEHUWKDWRQHRIWKHPDMRUUHDVRQV IRUXVLQJ*/0LVWKDWWKHUHVSRQVHYDULDQFHLVQRWFRQVWDQW$UHDVRQDEOH DVVXPSWLRQWRPDNHLVWKDWWKHYDULDQFHFKDQJHVZLWKWKHPHDQZKLFKLV W\SLFDO WR PRVW QRQQRUPDO GLVWULEXWLRQV )RU H[DPSOH IRU WKH H[SRQHQWLDOZHKDYHσ=µ. $VDUHVXOWVHOHFWLQJWKHUHVSRQVHGLVWULEXWLRQ DOVR GHWHUPLQHV WKH ZHLJKWV JLYHQ WR WKH YDULRXV REVHUYDWLRQV LQ WKH LWHUDWLYHO\UHZHLJKWHGOHDVWVTXDUHVSURFHGXUH 7KLV SURFHGXUH LV XVHG LQ */0 WR HVWLPDWH WKH PRGHO V SDUDPHWHUV DQGLWPLQLPL]HVWKHREMHFWLYHIXQFWLRQ QLVWKHVDPSOHVL]H
6
Q
\ L
L
L ! L
6LQFH WKH PHDQ GHSHQGV RQ WKH OLQHDU SUHGLFWRU WKURXJK WKH UHODWLRQVKLSµ J η), DQGVLQFHWKHYDULDQFHσGHSHQGVRQWKHPHDQLW LV REYLRXV WKDW ERWK WKH PHDQ DQG WKH YDULDQFH GHSHQG RQ WKH OLQHDU SUHGLFWRUWKHFRHIILFLHQWVRIZKLFKZHZLVKWRHVWLPDWHYLDPLQLPL]DWLRQ RI 7KLV H[SODLQV ZK\ DQ LWHUDWLYH SURFHGXUH LV QHHGHG 7KH ZHLJKWVσ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
&+$37(5$3352$&+(67202'(/6<67(0$7,&9$5,$7,21
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
5(63216(02'(/,1*0(7+2'2/2*<
VTXDUHV ZLWK LWV YDULRXV YDULDWLRQV WKH VHFRQG FRPSRQHQW GLIIHUV DSSUHFLDEO\ EHWZHHQ WKH DSSURDFKHV ,Q WKH UHVSRQVHWUDQVIRUPDWLRQ DSSURDFKWKHUHVSRQVHLVUHODWHGWRWKH/3YLDDQRQOLQHDUUHODWLRQVKLS FRQYH\HGE\DGDWDWUDQVIRUPDWLRQ ,QWKHFDVHRIWKH%&WUDQVIRUPDWLRQ YLD WKH SDUDPHWHU λ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
&+$37(5$3352$&+(67202'(/6<67(0$7,&9$5,$7,21
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
This page intentionally left blank
&KDSWHU
$SSURDFKHVWR0RGHO5DQGRP9DULDWLRQ
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λ
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
7KHUHDUHIRXUFDWHJRULHVRIJHQHUDODSSURDFKHVWRHPSLULFDOO\PRGHO UDQGRP YDULDWLRQ (DFK FDWHJRU\ HPDQDWHV IURP D GLIIHUHQW VHW RI DVVXPSWLRQVZKLFKSURYLGHVWKHWKHRUHWLFDOEDVLVIRULWVLPSOHPHQWDWLRQ 7KHWUXHGLVWULEXWLRQLVIURPDVSHFLILHGIDPLO\RIGLVWULEXWLRQV 7KLVDVVXPSWLRQUHQGHUVWKHDSSURDFKHPSLULFDOLQWKHVHQVHWKDWZHGR QRW DWWHPSW WR HVWDEOLVK WKDW WKH DVVXPSWLRQ LV WUXH EXW UHO\ RQ WKH JHQHUDOSUHYDOHQFHRIWKHDVVXPHGIDPLO\RIGLVWULEXWLRQVDVMXVWLILFDWLRQ WRILWWKLVIDPLO\WRWKHGDWDRQKDQG1RVWDWLVWLFDOWHVWLQJLVFDUULHGRXW WRHVWDEOLVKWKHFRUUHFWQHVVRIRXUDVVXPSWLRQ $QH[DPSOHLVWKH3HDUVRQIDPLO\RIGLVWULEXWLRQVZKLFKLQFOXGHVWKH EHWDJDPPDQRUPDO6WXGHQW VWDQG)GLVWULEXWLRQVDVVSHFLDOFDVHV,I WKLVIDPLO\LVVHOHFWHGIRUPRGHOLQJWKHQDOOZHKDYHWRGRLVHVWLPDWHLWV SDUDPHWHUV 7KH SUREDELOLW\ ZRXOG EH KLJK WKDW WKH WUXH XQGHUO\LQJ GLVWULEXWLRQ LQGHHG EHORQJV WR WKLV IDPLO\ RI GLVWULEXWLRQV &OHPHQWV SURFHGXUH IRU SURFHVV FDSDELOLW\ DQDO\VLV LQ D QRQQRUPDO HQYLURQPHQW UHODWHGWRLQ&KDSWHULQHVVHQFHIROORZVWKHVDPHORJLF 0RGHOLQJYLDDSDUDPHWHUULFKIDPLO\RIGLVWULEXWLRQV 7KH PDLQ FRQVLGHUDWLRQ KHUH LV WKDW WKH IDPLO\ RI GLVWULEXWLRQV VHOHFWHG IRUPRGHOLQJKDVHQRXJKSDUDPHWHUVWRUHWDLQWKHIOH[LELOLW\QHFHVVDU\WR PRGHOGLIIHUHQWO\VKDSHGGLVWULEXWLRQV8QOLNHWKHSUHYLRXVDSSURDFKWKH TXHVWLRQRIZKLFKIDPLO\LVDFWXDOO\VHOHFWHGIRUPRGHOLQJLVVHFRQGDU\ %\ HPSOR\LQJ D SDUDPHWHUULFK IDPLO\ RI GLVWULEXWLRQV LW LV JXDUDQWHHG WKDWVRPHRILWVPRPHQWVPDWFKWKRVHRIWKHXQGHUO\LQJ XQNQRZQ WUXH GLVWULEXWLRQ 7KLV LV DFFRPSOLVKHG E\ 0RPHQW PDWFKLQJ ZKHUH WKH SDUDPHWHUVRIWKHDGRSWHGGLVWULEXWLRQDUHGHWHUPLQHGVRWKDWWKHILUVW IHZ PRPHQWV ZLOO EH HTXDO PDWFK WKRVH RI WKH WUXH XQNQRZQ GLVWULEXWLRQ 7KH WUXH PRPHQWV DUH HVWLPDWHG IURP DYDLODEOH GDWD ,Q 6HFWLRQ ZH GLVFXVV ZK\ HTXDOLW\ RI PRPHQWV SURYLGHV JRRG DVVXUDQFHWKDWWKHXQGHUO\LQJWUXHGLVWULEXWLRQLVPRGHOHGZHOO:HZLOO DOVRGLVFXVVZK\LQSUDFWLFHWKLVDSSURDFKVXIIHUVVHULRXVGUDZEDFNV 1RWHWKDWWKLVDSSURDFKLVQRWDOWRJHWKHULQGHSHQGHQWRIWKHSUHYLRXV RQH 0RVWJHQHUDOV\VWHPVRIGLVWULEXWLRQVDOVRWHQGWREHSDUDPHWHUULFK
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
SXUVXHG ZKHQ PRPHQWPDWFKLQJ SURFHGXUHV IRU WKH 500 PRGHO DUH ODWHUGHYHORSHG 3DUDPHWHU5LFK)DPLOLHVRI'LVWULEXWLRQV7UDQVIRUPDWLRQVDQG ([SDQVLRQV :H VXUYH\ LQ WKLV VHFWLRQ VRPH FRPPRQO\ DSSOLHG JHQHUDO IDPLOLHV RI GLVWULEXWLRQV 7KH WHUP JHQHUDO UHIHUV WR WZR DVSHFWV DVVRFLDWHG ZLWK WKHVH IDPLOLHV )LUVW WKH IDPLOLHV LQFOXGH DV VSHFLDO FDVHV VRPH RI WKH PRVW ZLGHO\ XVHG GLVWULEXWLRQV 6HFRQGO\ WKHVH IDPLOLHV KDYH EHHQ VKRZQ WR EH IOH[LEOH HQRXJK WR RFFXS\ ODUJH SRUWLRQV RI WKH ^√β1, β2` SODQH:KLOHWKHODWWHULVGHVFULEHGLQGHWDLOODWHURQZKHQZHGHOLQHDWH GHVLUDEOH SURSHUWLHV RI HPSLULFDO PRGHOLQJ UHIHU WR 5HTXLUHPHQW LQ &KDSWHU LW LV SHUKDSV DSSURSULDWH WKDW WKH ^√β1, β2` SODQH EH EULHIO\ H[SODLQHGDWWKLVSRLQW 7KH GLVWULEXWLRQDO SDUDPHWHUV √β1 DQG β2 UHIHU WR PHDVXUHV RI WKH VKDSH RI WKH GLVWULEXWLRQ 7ZR IHDWXUHV RI WKLV VKDSH LV VNHZQHVV 6N ZKLFK UHODWHV WR KRZ DV\PPHWULF VNHZHG WKH GLVWULEXWLRQ LV DQG NXUWRVLV .X ZKLFKUHODWHVWRKRZORQJWKHWDLOVRIWKHGLVWULEXWLRQDUH DQGFRQVHTXHQWO\KRZQRQFRQFHQWUDWHGLWLVDURXQGLWVPHDQ6NHZQHVV LVPHDVXUHGE\√β1DQGWKHNXUWRVLVLVPHDVXUHGE\β2 RUβ2−3).)RUWKH QRUPDO GLVWULEXWLRQ ZH KDYH √β1= β2 )RU GLVWULEXWLRQV VNHZHG WR WKHULJKW ORQJULJKWWDLO ZHKDYH√β1!)RUGLVWULEXWLRQVZLWKORQJWDLOV ORQJHUWKDQWKHQRUPDO ZHKDYHβ2! ,QWHUPVRIFHQWUDOPRPHQWV PRPHQWVDURXQGWKHPHDQ {µN`DQG QRQFHQWUDOPRPHQWV PRPHQWVDERXW]HUR ^µN `VNHZQHVVDQGNXUWRVLV DUHFDOFXODWHGE\WKHIROORZLQJIRUPXODH
6N √β1 µσ3 [µ µ µ µ @σ3
.X β2 µσ4
>µ µ µ µ µ µ µ @σ4
ZKHUH µ µLVWKHPHDQµ ( < LVWKHQRQFHQWUDOVHFRQGPRPHQW DQGµ (> <µ @ σ2 LVWKHFHQWUDOVHFRQGPRPHQW WKHYDULDQFH )RUD V\PPHWULFDOGLVWULEXWLRQµ (> <µ @ .
5(63216(02'(/,1*0(7+2'2/2*<
:KHQ D JLYHQ SDUDPHWHUULFK IDPLO\ RI GLVWULEXWLRQV RFFXSLHV ODUJH SRUWLRQVRIWKH^√β1, β2`SODQHWKLVLPSOLHVWKDWE\DSURSHUFKRLFHRILWV SDUDPHWHUVWKLVIDPLO\RIGLVWULEXWLRQVFDQDWWDLQPRVWFRPELQDWLRQVRI VNHZQHVV DQG NXUWRVLV YDOXHV WKDW RQH PD\ HQFRXQWHU LQ SUDFWLFH )RU H[DPSOHIRUWKHH[SRQHQWLDOGLVWULEXWLRQZHKDYH√β1 β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
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
G)G\ > α\ ββ\β\ @ G)G\ ZKHUH)LVWKH&') ZLWKDUJXPHQW\ DQG³α´DQG³βL´ L DUH UHDOYDOXHGSDUDPHWHUV7KHVKDSHRIWKHGLVWULEXWLRQWKDWVDWLVILHV YDULHV FRQVLGHUDEO\ DFFRUGLQJ WR WKH YDOXHV RI WKH SDUDPHWHUV 3HDUVRQ FODVVLILHG WKHVH GLVWULEXWLRQV LQWR VHYHQ W\SHV HDFK IXOILOOLQJ FHUWDLQ FRQGLWLRQVZLWKUHVSHFWWRWKHSDUDPHWHUVβ0, β1 DQGβ2 7KH IROORZLQJ GLVWULEXWLRQV DUH PHPEHUV RI WKH 3HDUVRQ IDPLO\ D SDUWLDO OLVW 7KH QRUPDO GLVWULEXWLRQ WKH EHWD GLVWULEXWLRQ 6WXGHQW V W GLVWULEXWLRQ WKH JDPPD GLVWULEXWLRQ DQG WKH &DXFK\ GLVWULEXWLRQ $ JHQHUDO FODVVLILFDWLRQ RI 3HDUVRQ¶V GLVWULEXWLRQV LQWR VHYHQ FODVVHV LQFOXGLQJSDUDPHWHUFKDUDFWHUL]DWLRQ LVJLYHQLQ-.% 1RWHWKDW 3HDUVRQ WRRN DV D VWDUWLQJ SRLQW WKH VNHZHG ELQRPLDO DQG K\SHU JHRPHWULF GLVWULEXWLRQV ZKLFK KH VPRRWKHG LQ DQ DWWHPSW WR FRQVWUXFW VNHZHG FRQWLQXRXV GHQVLW\ IXQFWLRQV &RQWUDU\ WR WKH LPSUHVVLRQ RFFDVLRQDOO\FRQYH\HGE\PRGHUQZULWHUV3HDUVRQKLPVHOIGLGQRWEHJLQ KLVGHULYDWLRQRIWKH3HDUVRQV\VWHPZLWKWKHDERYHGLIIHUHQWLDOHTXDWLRQ 5RGULJXH]LQ.RW]DQG-RKQVRQ 9S
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
5(63216(02'(/,1*0(7+2'2/2*<
\IRUVRPHRIWKHVHIRUPVDUH ) \ >H[S \ @N
) \ >\F@N\!
) \ \F N\!
+HUHNDQGFDUHSRVLWLYHSDUDPHWHUV7KHVHIRUPVGHOLYHUVLPSOHLQYHUVH GLVWULEXWLRQIXQFWLRQV)RUH[DPSOHIRUWKHODVWIRUPZHREWDLQ \ > 3 N ±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±@JJ
<JK = J >H[S J= ±@H[S K= J
+HUH < LV WKH TXDQWLOH IXQFWLRQ H[SUHVVHG LQ WHUPV RI = D XQLW QRUPDO YDULDEOH DQG J DQG K DUH SDUDPHWHUV $OO WKHVH V\VWHPV DUH VKRZQ LQ &KDSWHUWREHVSHFLDOFDVHVRIWKH500PRGHO 5DPEHUJ DQG 6FKPHLVHU DQG 5DPEHUJ JHQHUDOL]HG 7XNH\ V WZRSDUDPHWHU /DPEGD GLVWULEXWLRQ LQWR D IRXUSDUDPHWHU JHQHUDOL]HG/DPEGDGLVWULEXWLRQ */' 7KHODWWHUKDVEHHQWKRURXJKO\ LQYHVWLJDWHG LQ .DULDQ DQG 'XGHZLF] .' ZKR KDYH GLVSDO\HG WKHTXDQWLOHIXQFWLRQRI*/'E\ \ 3 λ1 + λ2)>3λ3 3 λ4@3 ,QWKHLUERRN.'GHVFULEHYDULRXVSURFHGXUHVWRILW*/'WRJLYHQGDWD VHWV,Q&KDSWHUZHFRPSDUHDVROXWLRQREWDLQHGYLD*/' DQGWDNHQ
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
IURP.' WRDVROXWLRQGHULYHGE\ILWWLQJWKH500HUURUGLVWULEXWLRQ 6KRUH DE VXJJHVWHG VHYHUDO QHZ SDUDPHWHUULFK IDPLOLHV RI GLVWULEXWLRQVIRUHPSLULFDOPRGHOLQJRIUDQGRPYDULDWLRQ7KHVLPSOHVWLV #$#! #%&' & * ] K& -5#] 6# \ #$#! #%&' $ & * ] K& -5# ] 7 6 ZKHUH^]\`DUHWKHUHVSHFWLYH3WKTXDQWLOHVRIWKHVWDQGDUGQRUPDO= DQG WKH DSSUR[LPDWHG UY < ^µ, σ, 6N`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σ2 + µ2) 0 < >µ0 < @`> 0@
% >µ0 < $0@
$ ^0 < >0 < @`> 0@
% >0 < $0@
0 8
LV WKH ILUVW XSSHU SDUWLDO PRPHQW RI WKH VWDQGDUG
62
RESPONSE MODELING METHODOLOGY
normal distribution, M1(Y) and M2(Y) are partial upper moments of Y, namely,
M k (Y ) =
∫
∞
Med
y k f ( y)dy,
(5.13)
and Med is the median of Y. Note, that given a sample of n observations, an estimate for M1(Y) is the sum of all observations larger than the median, divided by n, and an estimate for M2(Y) is the sum of all squared observations (for observations larger than the median) divided again by n (not n/2!). For example, for the data: y={7,2,15,3}, an estimate of M2(Y) is M2(Y)= (72+152)/4= 68.5. Equation (5.11), with parameters given by (5.12), preserves the first two moments of Y (partial and complete). It has been shown to provide good representation to a wide variety of distributions, and it complies with the requirement that the MSEs of high degree moments, like skewness and kurtosis, are small when sample data are used to estimate its parameters (find details in Shore, 2000a, and also refer to Requirement 6 in Chapter 6). The piece-wise linear transformations, given in (5.8)-(5.12), are addressed in detail, with some numerical examples, in Chapter 21. A second parameter-rich family of distributions is (refer to Shore 2002a) A1 [P/(1-P)]B1 , P < 1/2, y= B2 A 2 {[P/(1-P)] - 1} + A1 , P ≥ 1/2.
(5.14)
This family of distributions looks similar to the generalized Lambda distribution (5.7). Fitting procedures were developed for (5.14) that again require estimates of only first and second degree moments, partial and complete (find details in Shore, 1998a,b, and also relate to Chapter 23). Another type of efforts to derive general families of distributions aimed at assembling under a single “umbrella distribution” several existing distributions. These endeavors have not introduced new
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
WKHRU\ EXW WULHG LQVWHDG WR UHIRUPXODWH NQRZQ GLVWULEXWLRQV LQ PRUH JHQHUDO WHUPV 1RWDEOH H[DPSOHV IRU WKLV DSSURDFK DUH WKH 'DUPRLV .RRSPDQFODVVRIGLVWULEXWLRQV WKHH[SRQHQWLDOW\SHFODVV SRZHUVHULHV GLVWULEXWLRQV WKH 3RO\DW\SH GLVWULEXWLRQV RU WKH FODVV RI VWDEOH GLVWULEXWLRQV UHIHUWR-.% IRUGHWDLOV 3HUKDSV WKH PRVW QRWDEOH RI WKHVH LV WKH H[SRQHQWLDO IDPLO\ RI GLVWULEXWLRQVFRPPRQO\LQWURGXFHGE\WKHGHQVLW\IXQFWLRQ
I \ H[S^>\θ−E θ @D φ F \φ `
ZKHUHD E DQGF DUHVSHFLILFIXQFWLRQVDQGθDQGφ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
5(63216(02'(/,1*0(7+2'2/2*<
7KH6/IDPLO\ IURPWKHORJQRUPDOGLVWULEXWLRQ
= γ + δORJ< <
= γ + δORJ>< < @<
7KH6%IDPLO\
7KH68IDPLO\
= γ + δ VLQK< γ + δ ORJ>< < @<
ZKHUH< ;ξ)/λ, ;LVWKHRULJLQDOUYWREHPRGHOHGYLDDPHPEHURI WKH-RKQVRQIDPLO\DQGξ DQGλ ! DUHORFDWLRQDQGVFDOHSDUDPHWHUV UHVSHFWLYHO\&RQYHQWLRQDOO\δ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
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
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
66
RESPONSE MODELING METHODOLOGY
that would still deliver a feeling for why moments determine a distribution. A well known statistical function that characterizes a distribution is the moment-generating function (MGF). It is defined by M(t) = E(etY) =
∫
∞
−∞
e ty dF ( y ) ,
(5.19)
where F(y) is the CDF of the r.v., Y. It can be shown mathematically that if the MGFs of two r.v.s, Y1 and Y2, are identical for all values of t in an interval around the point t=0, then the probability density functions of the two r.v.s must be identical. Using a well-known series expansion of ev, (5.20) ev = 1+v+v2/2!+v3/3!+.., we obtain from (5.19) M(t) = E[1+tY+(tY)2/2!+(tY)3/3!+..].
(5.21)
Although the series in (5.20) is infinite, it can be shown under fairly general conditions that the expected value of the sum is equal to the sum of the expected values (a result which is always true for a finite sum of r.v.s). This implies that one can write for (5.21) M(t) = 1+ tE(Y) + t2E(Y2)/2!+ t3E(Y3)/3!+.. . (5.22) Denoting the k-th non-central moment, E(Yk), by µk', we may re-write (5.22) by M(t) = 1+ t µ1' + (t2/2)µ2'+ (t3/6)µ3'+.. .
(5.23)
From (5.23) it can easily be recognized that the n-th derivative of M(t) with respect to t, at the point t=0, is equal to µn'. This is why M(t) is generally related to as "a moment-generating-function". The upshot of all these expressions is that since M(t) uniquely determines the distribution of the r.v., and since M(t) may be represented as a series expansion in terms of its moments, it is reasonable to assume that if the first few moments in this expansion are shared by two r.v.s their characteristic patterns of dispersion, as captured by the respective statistical distributions, will also tend to be similar. Staurt and Ord (1987, Section 3.34) provide a mathematical demonstration of this assertion, and numerous empirical studies repeatedly corroborated this fact.
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
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jYLVWKHWUXHGLVWULEXWLRQLVHVVHQWLDO$VDOOXGHGWRHDUOLHU6KHZKDUW FRQWURO FKDUWV XVH WKH QRUPDO GLVWULEXWLRQ IRU PRGHOLQJ DQG WKHUHIRUH RQO\ WKH ILUVW WZR PRPHQWV PHDQ DQG YDULDQFH DUH PDWFKHG ZKHQ D 6KHZKDUWFRQWUROFKDUWLVGHYHORSHG &RQYHUVHO\ &OHPHQWV GHYHORSHG D SURFHGXUH WR FDOFXODWH SURFHVV FDSDELOLW\ LQGLFHV IRU QRQQRUPDO SRSXODWLRQV EDVHG RQ WKH 3HDUVRQ IDPLO\ RI GLVWULEXWLRQV +LV SURFHGXUH UHTXLUHV PDWFKLQJ RI DOO
5(63216(02'(/,1*0(7+2'2/2*<
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³V\VWHPVRIIUHTXHQF\ FXUYHV´DOOKDYLQJWKHVDPHILUVWIRXUPRPHQWV,QVSHFWLQJWKHH[WHQVLYH WDEOHV JLYHQ WKHUHLQ RI YDULRXV SHUFHQWDJH SRLQWV RI GLIIHUHQW GLVWULEX WLRQV RQH PD\ DSSUHFLDWH WKH FORVHQHVV RI WKH YDOXHV RI UHVSHFWLYH SHUFHQWDJHSRLQWV TXDQWLOHYDOXHV GHULYHGIURPGLVWULEXWLRQVVKDULQJWKH VDPHILUVWIRXUPRPHQWV7KLVIDFWKDVOHGWRWKHXVHRIGLIIHUHQWIDPLOLHV RI GLVWULEXWLRQV OLNH -RKQVRQ IRU H[DPSOH )DUQXP 3HDUVRQ IRU H[DPSOH &OHPHQWV DQG RWKHUV ERWK LQ SURFHVV FDSDELOLW\
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
DQDO\VLV 3&$UHIHUWR.RW]DQG-RKQVRQ DQGLQVWDWLVWLFDOSURFHVV FRQWURO 63& :KDW GLVWULEXWLRQ LV YLUWXDOO\ XVHG IRU ILWWLQJ VHHPV WR KDYHDPDUJLQDOHIIHFW 7R VXP XS WKH EHVW JXDUDQWHH WKDW WKH WUXH GLVWULEXWLRQ KDV EHHQ SURSHUO\ PRGHOHG LV WR UHSODFH WKH XQNQRZQ GLVWULEXWLRQ E\ D NQRZQ IRXUSDUDPHWHU GLVWULEXWLRQ DQG WKHQ GHWHUPLQH WKH SDUDPHWHUV RI WKH ODWWHUVRWKDWHTXDOLW\RIWKHILUVWIRXUPRPHQWVLVDFKLHYHG :KLOH WKLV DVVHUWLRQ LV WKHRUHWLFDOO\ YDOLG WKHUH LV D SUREOHP KHUH 7KRXJKPDWFKLQJRIWKHILUVWIRXUPRPHQWVLVDGHVLUDEOHUHVXOWZHFDQ UDUHO\DFKLHYHLWLQSUDFWLFH7KLVLVGXHWRWKHIDFWWKDWVDPSOHHVWLPDWHV RI WKLUG DQG IRXUWK PRPHQWV DUH QRWRULRXVO\ NQRZQ IRU WKHLU ODUJH VDPSOLQJ HUURUV 8VLQJ VDPSOH HVWLPDWHV RI VNHZQHVV DQG NXUWRVLV LQ D ILWWLQJ SURFHGXUH PD\ OHDG WR H[WUHPHO\ ELDVHG HVWLPDWHV RI YDULRXV TXDQLOHV WKDW DUH FDOFXODWHG IURP WKH ILWWHG GLVWULEXWLRQ OLNH WKRVH XVHG IRU 3&$ .RW] DQG -RKQVRQ DQG .RW] DQG /RYHODFH UHSHDWHGO\ ZDUQ DJDLQVW WKLV HYHQWXDOLW\ LQ WKHLU UHIHUHQFH WR 3&$ IRU VNHZHGSRSXODWLRQV7KLVLVDOVRDPDMRULPSHGLPHQWWRWKHDSSOLFDWLRQ RI&OHPHQWVPHWKRGZKLFKLVWRGD\WKHPRVWZLGHO\DSSOLHGSURFHGXUH IRU3&$LQDQRQQRUPDOHQYLURQPHQW 8QIRUWXQDWHO\ XVHUV RIWKLV PHWKRG DUH QHYHU SUHZDUQHG DERXW WKLV GUDZEDFN LQKHUHQW WR DQ\ 3&$ ZKLFK UHOLHV RQ VDPSOH HVWLPDWHV RI VNHZQHVVRUNXUWRVLV /LNHZLVH.DULDQ DQG 'XGHZLF] .' DGYRFDWHWKH XVH RI SHUFHQWLOHPDWFKLQJ UDWKHU WKDQ PRPHQWPDWFKLQJ EHFDXVH WKH UHODWLYHO\ODUJHYDULDELOLW\RIVDPSOHPRPHQWVRIRUGHUDQGFDQPDNH LWGLIILFXOWWRREWDLQDFFXUDWHILWVWKURXJKWKHPHWKRGRIPRPHQWV .' S 6KRUH DE DOVR PDNHV UHSHDWHG UHIHUHQFH WR WKLVSRLQW,QSDUWLFXODUWKHODUJHPHDQVTXDUHGHUURUV 06(V WKDWUHVXOW IURP XVLQJ HVWLPDWHV RI KLJK GHJUHH PRPHQWV ZHUH GHPRQVWUDWHG YLD 0RQWH&DUORVLPXODWLRQZLWKUHJDUGWR3&$ 6KRUH E 7KHPDLQFRQFOXVLRQLVWKDWWKHPRVWLPSRUWDQWFULWHULRQWRMXGJHWKH PHULWRIDQHZDSSURDFKWRGHDOZLWKQRQQRUPDOLW\LQWKHIUDPHZRUNRI HPSLULFDOPRGHOLQJLVWKHGHJUHHWRZKLFKWKHILUVWWKUHHRUIRXUPRPHQWV RI WKH XQGHUO\LQJ GLVWULEXWLRQ DUH SUHVHUYHG E\ WKH GLVWULEXWLRQ WKDW DSSUR[LPDWHVLW,IVDPSOHHVWLPDWHVDUHXVHGWKLVFULWHULRQUHGXFHVWRWKH
5(63216(02'(/,1*0(7+2'2/2*<
TXHVWLRQRIWKHVL]HRIWKH06(VRIPRPHQWVDVVRFLDWHGZLWKWKHILWWHG HVWLPDWHG GLVWULEXWLRQ,QSUDFWLFHWKLVFULWHULRQLPSOLHVWKDWLIDODUJH QXPEHU RI VDPSOHV KDYH EHHQ WDNHQ IURP WKH VDPH QRQQRUPDO SRSXODWLRQDIRXUSDUDPHWHUGLVWULEXWLRQILWWHGIRUHDFKVDPSOH E\DQ\ VSHFLILHGPHWKRG DQGPRPHQWVFDOFXODWHG QXPHULFDOO\ IURPWKHILWWHG GLVWULEXWLRQWKHQWKH06(VRIWKHVHPRPHQWVVKRXOGEHPLQLPDO 7KH 06( FULWHULRQ VKRXOG DOVR EH WKH OHDGLQJ LQGLFDWRU LQ FRPSDUL VRQV EHWZHHQ GLIIHUHQW DSSURDFKHV 5HJUHWWDEO\ WKLV LV UDUHO\ WKH FDVH $V D EULHI UHYLHZ RI VRPH RI WKH UHIHUHQFHV PHQWLRQHG HDUOLHU ZRXOG DWWHVWFRPSDULVRQEHWZHHQPHWKRGVEDVHGRQWKH06(VRIPRPHQWVRI WKH ILWWHG GLVWULEXWLRQ LV UDUHO\ SUDFWLFHG DQG FRQVLGHUDWLRQV RI 06(V RIPRPHQWVDUHRIWHQPLVVLQJLQWKHOLWHUDWXUH)XUWKHUPRUHWKHZLGHXVH RI D IRXUPRPHQW ILWWLQJ SURFHGXUH LQ UHSRUWHG TXDOLW\ DQG UHOLDELOLW\ DSSOLFDWLRQV SURYLGH DPSOH HYLGHQFH WKDW WKLV FRQVLGHUDWLRQ LV UDUHO\ JLYHQWKHZHLJKWWKDWLWGHVHUYHV $OWHUQDWLYHO\PDWFKLQJSURFHGXUHVKDYHEHHQGHYHORSHGZKHUHERWK SDUWLDODQGFRPSOHWHPRPHQWVWDNHSDUWLQWKHILWWLQJSURFHGXUH
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
MXVWLILHGDQGWKHUHZDVQRQHHGWRPRGHOWKHUHDOSURFHVVGLVWULEXWLRQ :LWKWKHJUDGXDOWUDQVLWLRQWRSURFHVVPRQLWRULQJLQUHDOWLPHZKHUH LQGLYLGXDO REVHUYDWLRQV DQG QRW DYHUDJHV DUH SORWWHG LQ WKH FRQWURO FKDUWV WKH UHTXLUHPHQW IRU SURFHGXUHV WKDW ZRXOG UHIOHFW LQ WKH FRQWURO OLPLWVWKHWUXHSURFHVVGLVWULEXWLRQ RIWHQDQRQV\PPHWULFDORQH EHFDPH FUXFLDO&RQWUROFKDUWVZLWKDV\PPHWULFDOFRQWUROOLPLWVZHUHQHHGHG
)L 3U <\L
\L \
λ\Hλ\\≥
7KLVYDOXHRI)LPD\EHWUDQVIRUPHGLQWRVWDQGDUGQRUPDOTXDQWLOHYDOXH YLDWKH4VWDWLVWLF 4L Φ )L
5(63216(02'(/,1*0(7+2'2/2*<
ZKHUH Φ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«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σ/: IRUWKHORZHUSDUW RIWKHGLVWULEXWLRQDQGσ8:IRUWKHXSSHUSDUWZKHUH σ/: 3< σ<
σ8: 3<σ<
3< 3U <µ<
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
DQG ^µ< σ<`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
5(63216(02'(/,1*0(7+2'2/2*<
WKH PRPHQWV RI GLVWULEXWLRQV WKXV ILWWHG DUH VPDOOHU WKDQ WKH UHVSHFWLYH 06(VDVVRFLDWHGZLWKGLVWULEXWLRQVILWWHGE\PDWFKLQJRIGLUHFWVDPSOH HVWLPDWHV RI WKH ILUVW IRXU PRPHQWV ,Q IDFW LW KDV EHHQ VKRZQ 6KRUH WKDWWKLV ILWWLQJ SURFHGXUH SURYLGHV D EHWWHU HVWLPDWH RI VNHZQHVV GHULYHG E\ QXPHULFDO LQWHJUDWLRQ IURP WKH ILWWHG GLVWULEXWLRQ WKDQ WKDW GHULYHG IURP D GLUHFW VDPSOH HVWLPDWH RI VNHZQHVV 7KLV LPSOLHV WKDW EHWWHU UHSUHVHQWDWLRQ RI WKH XQNQRZQ GLVWULEXWLRQ LV DFFRPSOLVKHG UHODWLYHWRIRXUPRPHQWILWWLQJLUUHVSHFWLYHRIZKLFKGLVWULEXWLRQLVXVHG IRUWKHILWWLQJ )RUODUJHVDPSOHVL]HWKLVUHODWLYHDGYDQWDJHQDWXUDOO\YDQLVKHV :KHQ ILWWLQJ SURFHGXUHV DUH GHYHORSHG IRU WKH 500 HUURU GLVWUL EXWLRQ LQ &KDSWHU DQG IRU WKHLQYHUVH QRUPDOL]LQJ WUDQVIRUPDWLRQV LQ &KDSWHU RQO\ PRPHQWV RI VHFRQG GHJUHH DW PRVW SDUWLDO DQG FRPSOHWH DUH HPSOR\HG 3HUFHQWLOHEDVHG ILWWLQJ SURFHGXUHV DUH DOVR GHYHORSHG 5HIHUHQFHV >@$OEHUW-'HODPSDG\03RODVHN: $FODVVRIGLVWULEXWLRQVIRUUREXVWQHVV VWXGLHV-RXUQDORI6WDWLVWLFDO3ODQQLQJDQG,QIHUHQFH >@%DL'6&KRL,6 ;DQGUFRQWUROFKDUWVIRUVNHZHGSRSXODWLRQV-RXUQDO RI4XDOLW\7HFKQRORJ\ >@%R[*(3&R['5 $QDQDO\VLVRIWUDQVIRUPDWLRQV-RXUQDORIWKH5R\DO 6WDWLVWLFDO6RFLHW\6HULHV% >@ %XUU , : &XPXODWLYH IUHTXHQF\ IXQFWLRQV $QQDOV RI 0DWKHPDWLFDO 6WDWLVWLFV >@&KDQJ<6%DL'6 &RQWUROFKDUWVIRUSRVLWLYHO\VNHZHGSRSXODWLRQVZLWK ZHLJKWHG VWDQGDUG GHYLDWLRQV 4XDOLW\ DQG 5HOLDELOLW\ (QJLQHHULQJ ,QWHUQDWLRQDO >@ &KDQJ < 6 &KRL , 6 %DL ' 6 3URFHVV FDSDELOLW\ LQGLFHV IRU VNHZHG SRSXODWLRQV4XDOLW\DQG5HOLDELOLW\(QJLQHHULQJ,QWHUQDWLRQDO >@ &KRRELQHK ) %DOODUG - / &RQWURO OLPLWV RI 4& FKDUWV IRU VNHZHG GLVWULEXWLRQVXVLQJZHLJKWHGYDULDQFH,(((7UDQVDFWLRQVRQ5HOLDELOLW\ >@ &KRRELQHK ) %UDQWLQJ ' $ VLPSOH DSSUR[LPDWLRQ IRU VHPLYDULDQFH (XURSHDQ-RXUQDORI2SHUDWLRQDO5HVHDUFK > @&KRX<03RODQVN\$00DVRQ5/ 7UDQVIRUPLQJQRQQRUPDOGDWDWR QRUPDOLW\LQVWDWLVWLFDOSURFHVVFRQWURO-RXUQDORI4XDOLW\7HFKQRORJ\
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
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
5(63216(02'(/,1*0(7+2'2/2*<
>@4XHVHQEHUU\&3 E 63&4FKDUWVIRUD3RLVVRQSDUDPHWHUλ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¶V $GYDQFHG 7KHRU\ RI 6WDWLVWLFV 9 'LVWULEXWLRQ7KHRU\&KDUOHV*ULIILQ &RPSDQ\/WG/RQGRQ >@7DGLNDPDOOD35-RKQVRQ1/ 6\VWHPVRI IUHTXHQF\FXUYHVJHQHUDWHG E\WUDQVIRUPDWLRQVRIORJLVWLFYDULDEOHV%LRPHWULND >@7XNH\-: 7KHIXWXUHRIGDWDDQDO\VLV$QQDOVRI0DWKHPDWLFDO6WDWLVWLFV >@7XNH\-: 0RGHUQ7HFKQLTXHVLQ'DWD$QDO\VLV16)VSRQVRUHGUHJLRQDO UHVHDUFK FRQIHUHQFH DW 6RXWKHDVWHUQ 0DVVDFKXVHWWV 8QLYHUVLW\ 1RUWK 'DUPRXWK 0$
&+$37(5$3352$&+(67202'(/5$1'209$5,$7,21
>@ :X + 6 6ZDLQ - - )DUULQJWRQ 3 $ 0HVVLPHU 6 / $ ZHLJKWHG YDULDQFH FDSDELOLW\ LQGH[ IRU JHQHUDO QRQQRUPDO SURFHVV 4XDOLW\ DQG 5HOLDELOLW\ (QJLQHHULQJ,QWHUQDWLRQDO > @9RLW(2 7KH6GLVWULEXWLRQ $WRROIRUDSSUR[LPDWLRQDQGFODVVLILFDWLRQRI XQLYDULDWHXQLPRGDOSUREDELOLW\GLVWULEXWLRQV%LRPHWULFDO-RXUQDO
This page intentionally left blank
&KDSWHU
7KH5HTXLUHPHQWVDQG(YDOXDWLRQRI &RPSOLDQFH
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
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(55(48,5(0(176$1'(9$/8$7,212)&203/,$1&(
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
5(63216(02'(/,1*0(7+2'2/2*<
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³/LQHDU3UHGLFWRU´ /3 &RPPHQW )RU EUHYLW\ ZH ZLOO KHQFHIRUWK UHODWH RQO\ WR FRQYH[ UHODWLRQVKLSV XQGHUVWDQGLQJ WKDW VLPLODU DVVHUWLRQV PD\ EH PDGH ZLWK UHVSHFWWRFRQFDYHUHODWLRQVKLSV 5HDVRQLQJ7KHUHTXLUHPHQWUHODWHVWRWKHHPSLULFDOREVHUYDWLRQWKDW D ODUJH SRUWLRQ RI UHODWLRQVKLSV REVHUYHG LQ QDWXUH DQG UHIOHFWHG LQ FRUUHVSRQGLQJ VFLHQWLILF DQG HQJLQHHULQJ PRGHOV LV FRPSULVHG RI PRQRWRQHFRQYH[UHODWLRQVKLSV UHIHUWRWKHVXUYH\LQ&KDSWHUDQGDOVR WR6KRUHD $OOOLQNIXQFWLRQVXVHGLQWKHIUDPHZRUNRI*/0DUH OLNHZLVHPRQRWRQHFRQYH[IXQFWLRQV([DPSOHVIRUFRQYH[IXQFWLRQVDUH ;DH[DQGH[S [D
&+$37(55(48,5(0(176$1'(9$/8$7,212)&203/,$1&(
$Q ([DPSOH &RQVLGHU WKH %& QRUPDOL]LQJ WUDQVIRUPDWLRQ %R[ DQG &R[
< E $ F < ORJ < E $ F
ZKHUH η= β0+β1;β;βN;N LV WKH /3 ε LV WKH HUURU SUREDEO\ QRUPDO DQG^;L`DUHWKHHIIHFWV SUHGLFWRUYDULDEOHV LQFOXGHGLQWKH/3 1RWHWKDWWKHODWWHUPD\FRQWDLQQRQOLQHDUWHUPVDQGWKHOLQHDUUHIHU DV LQ OLQHDU UHJUHVVLRQ WR OLQHDULW\ LQ WKH SDUDPHWHUV RQO\ /LNHZLVH FRQYH[LW\UHODWHVWRWKH/3DQGQRWQHFHVVDULO\WRLQGLYLGXDOIDFWRUV )URPUHODWLRQVKLS WKHQRQOLQHDUPRGHOIRU< ! LV
> E $ F $ @ < H[S E $ F
DQGLQWHUPVRIWKHVWDQGDUGQRUPDOYDULDEOH=
> E $ ! = $ @ < H[S E $ ! =
ZKHUH σ LV WKH HUURU VWDQGDUG GHYLDWLRQ (TXDWLRQV DQG DUH SRZHUDQGH[SRQHQWLDOUHODWLRQVKLSV GHSHQGHQWRQWKHYDOXHRIλ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
5(63216(02'(/,1*0(7+2'2/2*<
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λZKLFKGHWHUPLQHV WKHQRQOLQHDUVWUXFWXUHRIWKHPRGHOLVGHFLGHGVROHO\E\GDWDDQDO\VLV LQFRPSOLDQFHZLWK5HTXLUHPHQW 5HTXLUHPHQW 3URYLGHOLQHDUUHODWLRQVKLSDVDVSHFLDOFDVH 'HVFULSWLRQ $Q HPSLULFDO PRGHOLQJ PHWKRGRORJ\ VKRXOG DFFRPPRGDWH OLQHDU UHODWLRQVKLS WKH UHVSRQVH LV HTXDO WR WKH OLQHDU SUHGLFWRUSOXVDQHUURUWHUP 5HDVRQLQJ7KH FRPPRQ SUHYDOHQFHRIOLQHDU PRGHOV DVMXGJHG E\ WKH ZLGH DQG HIIHFWLYH XVH RI PXOWLSOH OLQHDU UHJUHVVLRQ DQDO\VLV LQ HPSLULFDO PRGHOLQJ UHQGHUV WKLV SURSHUW\ HVVHQWLDO WR DQ\ JHQHUDO PHWKRGRORJ\IRUHPSLULFDOPRGHOLQJ $Q ([DPSOH &RQVLGHU WKH %& WUDQVIRUPDWLRQ )RU λ ZH REWDLQDOLQHDUUHODWLRQVKLSEHWZHHQηDQGWKHUHVSRQVH5HTXLUHPHQWLV IXOILOOHGE\WKH%&WUDQVIRUPDWLRQ 5HTXLUHPHQW $GXDOHUURUVWUXFWXUH 'HVFULSWLRQ 7KLV SURSHUW\ LPSOLHV WKDW DQ HPSLULFDO PRGHO VKRXOG LQFOXGH WZR HUURU WHUPV SRVVLEO\ FRUUHODWHG $Q HUURU WHUP WKDW LV DVVRFLDWHG ZLWK WKH /3 DQG DQRWKHU HUURU WHUP GLUHFWO\ DVVRFLDWHG ZLWK WKHUHVSRQVH 5HDVRQLQJ7KLVSURSHUW\LVDORJLFDOFRQFOXVLRQRIWKHGHILQLWLRQRID UHODWLRQDOPRGHO)RUDQ\VXFKPRGHORQHPD\SODXVLEO\FRQFHLYHRID
&+$37(55(48,5(0(176$1'(9$/8$7,212)&203/,$1&(
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β0 + β1; β2; WKHQ REYLRXVO\ WKH DGGLWLYH HUURU WHUP VKRXOG EH FRQVLGHUHG DV DQ DSSUR[LPDWLRQWRWKHWUXHHUURU7KXVLIZHZULWH; µ+ε WKHPHDQSOXV HUURU WKHQWKHDERYHHTXDWLRQEHFRPHV β0 + β1(µ+ε) + β2(µ+ε)2 = β0 + β1µ + β2µ2 + (β1+ 2β2µ)ε + β2ε2. 7KH ODVW WHUP LV QHJOLJLEOH DV ZH PD\ DVVXPH WKDW ε2<<ε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
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(55(48,5(0(176$1'(9$/8$7,212)&203/,$1&(
: ORJ < α/λ)[(η+ε1)λ−1] + µ2 + ε2
ZKHUH ε1 DQG ε2 ҏDUH LQGHSHQGHQW QRUPDO HUURUV ZLWK ]HUR PHDQ DQG VWDQGDUGGHYLDWLRQVσε1DQGσε2UHVSHFWLYHO\7KHGHQVLW\IXQFWLRQRI: KDVEHHQGHYHORSHG 6KRUH DQGVKRZQWREHHLWKHUQRQQRUPDORU QRUPDO GHSHQGHQW RQ LWV SDUDPHWHUV ,W PD\ EH HDVLO\ UHDOL]HG IURP WKDWDVηLQFUHDVHVε1ҏ JUDGXDOO\EHFRPHVQHJOLJLEOH ε1<<η 7KLV LPSOLHVWKDWZLWKDSSUHFLDEOHV\VWHPDWLFYDULDWLRQLQWKHOLQHDUSUHGLFWRU η WKHHUURUGLVWULEXWLRQRI:ZLOOQRWEHFRQVWDQW$Vη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¶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
5(63216(02'(/,1*0(7+2'2/2*<
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η WRREWDLQIURP
> $ $ ! = @ < H[S $ ! =
7KLV HTXDWLRQ H[SUHVVHV WKH UDQGRP YDULDEOH < LQ WHUPV RI D VWDQGDUG QRUPDO YDULDEOH = ,Q IDFW IRU DQ\ VSHFLILHG TXDQWLOH RI WKH QRUPDOYDULDEOHWKHFRUUHVSRQGLQJTXDQWLOHRI<PD\EHFDOFXODWHGIURP )RU LQVWDQFH VHWWLQJ Ζ WKH UHVSRQVH PHGLDQ λ)1/λ λ LV REWDLQHG 7KH UHDGHU PD\ HDVLO\ GHGXFH WKDW WKH HUURU GLVWULEXWLRQ DVVRFLDWHGZLWKWKH%&WUDQVIRUPDWLRQFDQQRWDFFRPPRGDWHGLIIHUHQWO\ VKDSHGHUURUGLVWULEXWLRQVVLQFHDVLQJOHSDUDPHWHUλLVQRWVXIILFLHQWWR DFFRPPRGDWHGLIIHUHQWVHWVRIYDOXHVIRUWKHILUVWWKUHHRUIRXUPRPHQWV ,Q FRQFOXVLRQ WKH LQYHUVH %& WUDQVIRUPDWLRQ DV JLYHQ E\ LV LQDGHTXDWHIRUPRGHOLQJUDQGRPYDULDWLRQ 5HTXLUHPHQW 3URYLGHJRRGFRYHUDJHRIWKH √ √β1, β2 SODQH 'HVFULSWLRQ √β1 DQG β2−3 DUH WKH WUDGLWLRQDO PHDVXUHV RI VNHZQHVV DQG NXUWRVLV FRUUHVSRQGLQJ UHVSHFWLYHO\ WR WKH WKLUG DQG IRXUWK FXPXODQWV RI WKH VWDQGDUGL]HG YDULDEOH )RU WKH QRUPDO GLVWULEXWLRQ
&+$37(55(48,5(0(176$1'(9$/8$7,212)&203/,$1&(
√β1= β2− )RU D GLVFXVVLRQ RI FXPXODQWV UHIHU WR 6WXDUW DQG 2UG 6HFWLRQ 7KH √β1, β2 SODQH ZDV LQWURGXFHG LQ &KDSWHU 5HTXLUHPHQWVWDWHVWKDWDPHWKRGRORJ\IRUHPSLULFDOPRGHOLQJVKRXOG EHVKRZQWRKDYHDQHUURUGLVWULEXWLRQWKDWRFFXSLHVDUHDVRIWKH √β1, β2 SODQH VLPLODU WR WKRVH RFFXSLHG E\ GLVWULEXWLRQV ZLGHO\ HQFRXQWHUHG LQ SUDFWLFH 5HDVRQLQJ,QHPSLULFDOPRGHOLQJRIUDQGRPYDULDWLRQLWLVVWDQGDUG SUDFWLFH WR GHPRQVWUDWH WKDW ZKHQ PRGHOLQJ LV LPSOHPHQWHG YLD VRPH SDUDPHWHUULFK IDPLO\ RI GLVWULEXWLRQV OLNH 3HDUVRQ RU WKH *HQHUDOL]HG /DPEGD WKH IDPLO\ XVHG IRU ILWWLQJ KDV PHPEHUV ZLWK √β1, β2) YDOXHV WKDWVSDQODUJHSRUWLRQVRIWKH √β1, β2 SODQH7KLVSURSHUW\UHIOHFWVWKH FDSDELOLW\RIWKHIDPLO\RIGLVWULEXWLRQVWRSURYLGHJRRGUHSUHVHQWDWLRQWR D ODUJH VSHFWUXP RI VKDSH FKDUDFWHULVWLFV DVVRFLDWHG ZLWK GLVWULEXWLRQV WKDWDSUDFWLWLRQHUPD\HQFRXQWHU /LNHZLVH WKH HUURU GLVWULEXWLRQ HPEHGGHG LQ D JHQHUDO HPSLULFDO PRGHO VKRXOG EH DEOH WR DFFRPPRGDWH ODUJH SRUWLRQV RI WKH √β1, β2) SODQH ,Q SUDFWLFDO WHUPV UHDOL]DWLRQ RI WKLV SURSHUW\ LV FRQGLWLRQDO RQ DFKLHYLQJ 5HTXLUHPHQW 2QH FDQQRW H[SHFW DQ HUURU GLVWULEXWLRQ WR SURYLGHJRRGFRYHUDJHRIWKH √β1, β2 SODQHXQOHVVLWKDVDVXIILFLHQWO\ ODUJH QXPEHU RI SDUDPHWHUV 7KH UHSUHVHQWDWLRQ IHDWXUH ZKLFK 5HTXLUHPHQW GHPDQGV LV FRQGLWLRQDO RQ KDYLQJ HQRXJK SDUDPHWHUV 5HTXLUHPHQW 7KH VWDWLVWLFDO OLWHUDWXUH VKRZV WKDW WKH ODWWHU LV QRW D VXIILFLHQW FRQGLWLRQ IRU WKH IXOILOOLQJ RI 5HTXLUHPHQW 7KH IROORZLQJ H[DPSOHGHPRQVWUDWHVWKLV $Q([DPSOH5HODWHWRWKHWKUHHSDUDPHWHUJDPPDGLVWULEXWLRQZLWK GHQVLW\IXQFWLRQ -RKQVRQ.RW]DQG%DODNULVKQDQ &KDSWHU
I \ \γ)(α−1) H[S> \γ)/β] / [βα Γ(α)], α > , β > \!γ
−1/2
)RUWKLV GLVWULEXWLRQ WKHVNHZQHVV PHDVXUH √βң LVHTXDO α DQG WKH NXUWRVLV PHDVXUH β2 LV HTXDO α−1 8VLQJ WKLV GLVWULEXWLRQ DV D SODWIRUP IRU HPSLULFDO PRGHOLQJ RI UDQGRP YDULDWLRQ 5HTXLUHPHQW LV QRWIXOILOOHG)LUVWVLQFHERWKVNHZQHVVDQGNXUWRVLVGHSHQGRQDVLQJOH SDUDPHWHUQRIOH[LELOLW\LVSUHVHUYHGWRPRGHOGLVWULEXWLRQVZLWKYDU\LQJ YDOXHVRIVNHZQHVVDQGNXUWRVLV6HFRQGO\VLQFHα>QHJDWLYHVNHZQHVV
5(63216(02'(/,1*0(7+2'2/2*<
ORQJOHIWWDLOHGGLVWULEXWLRQ FDQQRWEHDFKLHYHG+RZHYHU5HTXLUHPHQW LVIXOILOOHG>VLQFH KDVWKUHHSDUDPHWHUV@ 5HTXLUHPHQW 3DUVLPRQ\LQPRGHOLQJ 'HVFULSWLRQ /HW WKH PRGHO EH DV VLPSOH DV SRVVLEOH ,Q SUDFWLFDO WHUPVWKLVLPSOLHVVLPSOHVWVWUXFWXUHDQGDVPDOOQXPEHURISDUDPHWHUV WKDWQHHGVWREHHVWLPDWHG 5HDVRQLQJ7KLVSURSHUW\LVWKHXOWLPDWHJRDORIDQ\PRGHOLQJHIIRUW %R[ GHILQHVSDUVLPRQ\DV³WKHSURYLVLRQRIWKHVLPSOHVWDGGLWLYH PRGHOV´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
&+$37(55(48,5(0(176$1'(9$/8$7,212)&203/,$1&(
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
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(55(48,5(0(176$1'(9$/8$7,212)&203/,$1&(
7DEOH&RPSDULVRQRI&RPSOLDQFHZLWK5HTXLUHPHQWVRI0DMRU$SSURDFKHVIRU 0RGHOLQJ6\VWHPDWLF9DULDWLRQ 5HTXLUHPHQW &RQYH[LW\ $OO'DWD'ULYHQ ,QFO/LQHDU 'XDOHUURU &KDQJLQJHUURU GLVW 0RPHQWVSUHVHUYHG 5HSUHVHQWDELOLW\ 3DUVLPRQ\ &RPSDWZLWK FXUUHQWPHWKRG (DV\WRDSSO\
0HWKRGRORJ\ /LQHDUUHJUHVVLRQ %&7UDQVIRUPDWLRQ */0
500
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
5(63216(02'(/,1*0(7+2'2/2*<
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λ)5HTXLUHPHQWLVDOVRQRWVDWLVILHGVLQFHRQFHλ LVGHWHUPLQHGFKDQJHVLQWKHRUGHURIPDJQLWXGHRIWKHOLQHDUSUHGLFWRU GR QRW PRGLI\ WKH VKDSH FKDUDFWHULVWLFV RI WKH DVVRFLDWHG HUURU GLVWULEXWLRQ 7KLV DIIHFWV DOVR FRPSOLDQFH ZLWK 5HTXLUHPHQWV DQG EHFDXVH D VLQJOH VKDSH SDUDPHWHU λ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
&+$37(55(48,5(0(176$1'(9$/8$7,212)&203/,$1&(
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λ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
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(55(48,5(0(176$1'(9$/8$7,212)&203/,$1&(
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j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
5(63216(02'(/,1*0(7+2'2/2*<
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¶V $GYDQFHG 7KHRU\ RI 6WDWLVWLFV 9 'LVWULEXWLRQ7KHRU\&KDUOHV*ULIILQ &RPSDQ\/WG/RQGRQ
3DUW,,
500'(9(/23,1*$1' (9$/8$7,1*7+(*(1(5$/ $3352$&+
This page intentionally left blank
&KDSWHU
7KH5000RGHO
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
5(63216(02'(/,1*0(7+2'2/2*<
WKH ³/DGGHU RI IXQGDPHQWDO XQLIRUPO\ FRQYH[FRQFDYH IXQFWLRQV´ $OWKRXJKPRQRWRQHFRQYH[LW\KDGDGPLWWHGO\JXLGHGXVLQVHOHFWLQJWKH PRGHOVLQFOXGHGLQWKHVXUYH\RI&KDSWHULWZDVHYLGHQWIURPWKHZLGH VSHFWUXP RI PRGHOV UHYLHZHG WKDW XQLIRUPO\ FRQYH[FRQFDYH UHODWLRQVKLSVSHUPHDWHVFLHQWLILFDQGHQJLQHHULQJPRGHOLQJ ,Q &KDSWHUV DQG FXUUHQW JHQHUDO DSSURDFKHV WR HPSLULFDO PRGHOLQJ RI D UHVSRQVH YDULDWLRQ ZHUH EULHIO\ RXWOLQHG 7KHVH UHODWHG ERWK WR V\VWHPDWLF YDULDWLRQ &KDSWHU DQG WR UDQGRP YDULDWLRQ &KDSWHU &KDSWHU H[SRXQGHG GHVLUDEOH UHTXLUHPHQWV WKDW DQ\ JHQHUDOPHWKRGRORJ\IRUHPSLULFDOPRGHOLQJVKRXOGVDWLVI\$VVHVVLQJ WKH GHJUHH RI FRPSOLDQFH RI FXUUHQW PHWKRGRORJLHV ZLWK WKHVH UHTXLUHPHQWVDOORZHGDQHYDOXDWLRQRIWKHLUTXDOLW\,QKHUHQWGUDZEDFNV RI H[LVWLQJ PRGHOLQJ DSSURDFKHV ZHUH LQGLFDWHG ,Q SDUWLFXODU LW ZDV HPSKDVL]HG WKDW WKH PRGHOV GHULYHG LQ WKH IUDPHZRUN RI FXUUHQW DSSURDFKHVGRQRWUHDOO\FDSWXUHVRPHRIWKHIXQGDPHQWDOFKDUDFWHULVWLFV WKDW DUH FRPPRQO\ VKDUHG E\ PDLQVWUHDP PRGHOV GHYHORSHG LQGHSHQGHQWO\LQVFLHQWLILFDQGHQJLQHHULQJGLVFLSOLQHV 500LVDQDWWHPSWWRILOOLQWKHYRLGFUHDWHGE\WKHVKRUWFRPLQJVRI FXUUHQWPHWKRGRORJLHVIRUHPSLULFDOPRGHOLQJ ,Q6HFWLRQILYHDVVXPSWLRQVDUHPDGHZKLFKDOORZGHULYDWLRQRI WKH EDVLF 500 PRGHO LQ DQ D[LRPDWLF IDVKLRQ 7KLV PRGHO ZLOO ODWHU VHUYH DV WKH FRUH SODWIRUP IRU WKH GHYHORSPHQW RI WKH QHZ 5HVSRQVH 0RGHOLQJ 0HWKRGRORJ\ 500 1RWH WKDW WKH D[LRPDWLF GHULYDWLRQ GHWDLOHG KHUHZLWK LV VRPHZKDW GLIIHUHQW IURP WKH RULJLQDO GHULYDWLRQ 6KRUH LQ WKDW D PRUH JHQHUDO PRGHO LV DVVXPHG WKDW DOORZV WKH HUURUVWREHFRUUHODWHG ([SUHVVLRQVIRUWKHUHVSRQVHPRPHQWVDUHGHULYHGLQ6HFWLRQ7KH ODVW 6HFWLRQ H[SORUHV WKH UHODWLRQVKLS EHWZHHQ WKH FRHIILFLHQW RI YDULDWLRQ RI WKH 500 PRGHO DQG WKH OLQHDU SUHGLFWRU /3 ,W LV VKRZQ WKDWDV/3LQFUHDVHVWKHYDULDQFHWHQGVWRVWDELOL]H EHLQGHSHQGHQWRIWKH PHDQ ,Q&KDSWHUZHZLOOGHYHORSHVWLPDWLRQSURFHGXUHVDQGLQ&KDSWHU WKH500HUURUGLVWULEXWLRQZLOOEHGHULYHGDQGLWVSURSHUWLHVH[SORUHG
&+$37(57+(50002'(/
$Q$[LRPDWLF'HULYDWLRQRIWKH5000RGHO 'HYHORSLQJ WKH JHQHUDO PRGHO ZH DWWHPSW WR FDSWXUH WKH EHKDYLRU RI D UDQGRP UHVSRQVH ZKLFK H[SHULHQFHV WZR LQGHSHQGHQW VRXUFHV RI YDULDWLRQ ([WHUQDOO\JHQHUDWHG YDULDWLRQ DQG 6HOIJHQHUDWHG YDULDWLRQ 7KHIRUPHU YDULDWLRQ LV DVVXPHG WR FRPSULVH ERWKV\VWHPDWLF YDULDWLRQ WUDQVPLWWHGWRWKHUHVSRQVHYLDWKH/3 GHQRWHG η DQGUDQGRPYDULDWLRQ WUDQVPLWWHG WR WKH UHVSRQVH E\ DQ DGGLWLYH HUURU ε1. 6HOIJHQHUDWHG YDULDWLRQUHIOHFWVWKHLQQHUQDWXUDOLQVWDELOLW\RIWKHPRGHOHGUHVSRQVHDV ZHOODVPHDVXUHPHQWLPSUHFLVLRQ,WLVUHSUHVHQWHGE\DUDQGRPHUURUε2 7KHWZRVRXUFHVRIYDULDWLRQDUHDVVXPHGWRLQWHUDFWZLWKRQHDQRWKHU $OOWKHVHIHDWXUHVDUHVXPPDUL]HGLQDVHWRIILYHDVVXPSWLRQVZKLFK IRUPWKHD[LRPDWLFEDVLVIRUWKHGHYHORSPHQWRIWKHJHQHUDOPRGHO ,QWKHIROORZLQJVXEVHFWLRQVWKH500PRGHOLVGHYHORSHGVWDUWLQJ ZLWK WKH DVVXPSWLRQV 6XEVHFWLRQ DQG FRQFOXGLQJ ZLWK WKH PDWKHPDWLFDOH[SUHVVLRQWKDWH[SUHVVHVWKHUHVSRQVHLQWHUPVRIWKH/3 WKHUDQGRPHUURUVDQGWKHPRGHO VSDUDPHWHUV 6XEVHFWLRQ 7KHPRGHODVVXPSWLRQV
)LYHDVVXPSWLRQVSURGXFHWKHEDVLVIRUWKHQHZ500PRGHO
$VVXPSWLRQ $ 7KH PRGHOHG UHVSRQVH LV D QRQQHJDWLYH UY < ≥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
5(63216(02'(/,1*0(7+2'2/2*<
6HOIJHQHUDWHG GHYLDWLRQV DUH SURGXFHG E\ D UDQGRP PHFKDQLVP GULYHQE\DQRUPDOUDQGRPYDULDEOH $VVXPSWLRQ % 7KH PRGHOHG UHVSRQVH < ≥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ρ ρ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
&+$37(57+(50002'(/
([WHUQDOO\JHQHUDWHG YDULDWLRQ LV PRGHOHG E\ I η,ε θ1 ,Q WKLV H[SUHVVLRQηLVWKH/3DQGε1 = (σε1)Ζ1LVDQDGGLWLYHHUURUWHUPZKHUH = LV D VWDQGDUG QRUPDO YDULDEOH QDPHO\ WKH HUURU LV QRUPDOO\ GLVWULEXWHGZLWK]HURPHDQDQGVWDQGDUGGHYLDWLRQσε1). θ1LVDYHFWRURI SDUDPHWHUV WKDW QHHG WR EH GHWHUPLQHG HVWLPDWHG ,W LV DVVXPHG WKDW I η,ε θ1 LVPRQRWRQHFRQYH[RUFRQFDYH 6HOIJHQHUDWHG YDULDWLRQ LV PRGHOHG E\ I ε2; θ2 ,Q WKLV H[SUHVVLRQ ε2= (σε2)=LVDUDQGRPHUURU=LVDVWDQGDUGQRUPDOYDULDEOH QDPHO\ WKHHUURULVQRUPDOO\GLVWULEXWHGZLWK]HURPHDQDQGVWDQGDUGGHYLDWLRQ σε2 DQGθ θ2LVDYHFWRURISDUDPHWHUVWKDWQHHGWREHGHWHUPLQHG $VLPSOLHGE\DVVXPSWLRQ(WKHUHVSRQVHLVPRGHOHGDV < = I1(η, ε1; θ1) I2(ε2; θ2), (7.1) ZKHUHIDQGIDUHDVGHILQHGDERYHDQGWKHUDQGRPYDULDEOHV^==` DUHDVVXPHGWRGHULYHIURPDELYDULDWHVWDQGDUGQRUPDOGLVWULEXWLRQZLWK FRUUHODWLRQρ. 7KHIXQFWLRQVIDQGIZLOOEHGHYHORSHGLQWKHQH[WVXE VHFWLRQVDQGUHVSHFWLYHO\ 'HULYLQJI %\ $VVXPSWLRQ $ WKH VLPSOHVW PRGHO IRU VHOIJHQHUDWHG UDQGRP YDULDWLRQZKLFKLQFUHDVHVZLWK0LVDSURSRUWLRQDORQH I2(ε2; θ2) = Μ(1 + ε2) = Μ(1 + σε2Ζ2) = Μ + (Μσε2)Ζ2.
(7.2)
)RUWKHDEVROXWHYDOXHRIWKHUHODWLYHUDQGRPHUURUZHPD\DVVXPHWKDW ŇI2 − ΜŇ/ Μ = |ε2| << 1.
(7.3)
7KHUHIRUH PD\EHDSSUR[LPDWHO\UHSUHVHQWHGE\ I2(ε2; θ2) = Μ(1 + ε2) = Μ(1+ε2) (1/ε2) ε2 ≈ H[S>ORJ 0 ε@ H[S>µσε=@ѽҏ
(7.4)
ZKHUH θ2 = {µ2, σε` DQG µ DQG σε DUH WKH PHDQ DQG WKH VWDQGDUG GHYLDWLRQ RI WKH QRUPDO YDULDEOH WKDW JHQHUDWHV VHOIYDULDWLRQ ,Q RWKHU ZRUGV VHOIJHQHUDWHG YDULDWLRQ LV PRGHOHG E\ D ORJQRUPDO YDULDEOH D ORJQRUPDO YDULDEOH LV DQ\ UHVSRQVH ZKLFK LV QRUPDOO\ GLVWULEXWHG RQ D ORJWUDQVIRUPHGVFDOH $V LV HYLGHQW E\ WKH VXUYH\ RI FXUUHQW PDLQVWUHDP PRGHOV UHIHU WR &KDSWHUV DQG ERWK YHUVLRQV RI WKH HUURU DQG DUH TXLWH
5(63216(02'(/,1*0(7+2'2/2*<
FRPPRQ7KXV LVW\SLFDOO\DVVXPHGZKHQWKHORJRIWKHUHVSRQVHLV PRGHOHG ZKLOH LV DGRSWHG ZKHQ WKH UHVSRQVH LV PRGHOHG RQ WKH RULJLQDO VFDOH ,I ε2 IXOILOOV WKH WZR IRUPXODWLRQV DUH DERXW HTXLYDOHQW 'HULYLQJI %\ DVVXPSWLRQV % DQG ' WUDQVPLWWHG YDULDWLRQ LV PRGHOHG E\ WKH UHODWLRQVKLS EHWZHHQ WKH UHVSRQVH DQG WKH /3 ZLWK WKH DVVRFLDWHG UDQGRP HUURU DQG WKLV UHODWLRQVKLS LV PRQRWRQH FRQYH[ 2QH FDQ FRQFHLYH RI D P\ULDG RI IXQFWLRQV ZLWK WKLV FKDUDFWHUL]DWLRQ 7KHUHIRUH ZHVKRXOGDVNRXUVHOYHVZKHWKHUWKHXQLYHUVHRIDOOPRQRWRQHLQFUHDVLQJ FRQYH[IXQFWLRQVWKDWPD\SRWHQWLDOO\VHUYHWRPRGHOI η,ε θ1 FDQEH FDSWXUHGE\DOLPLWHGVHWRIVRPHIXQGDPHQWDOFRQYH[IXQFWLRQVZKLFK PD\ GHOLYHU WKH UHTXLUHG UHSUHVHQWDWLRQ IRU YDU\LQJ GHJUHHV RI FRQYH[LW\" ,Q IDFW ZH KDYH DOUHDG\ WDFNOHG WKLV SUREOHP LQ IRUPXODWLQJ 7KH ODGGHURIIXQGDPHQWDOXQLIRUPO\FRQYH[IXQFWLRQV &KDSWHU 7KHUHLQ LW ZDV HPSKDVL]HG WKDW JLYHQ WKH QXPEHU RI HQJLQHHULQJ DQG VFLHQWLILF PRGHOVWKDWDUHFRPSDWLEOHZLWK³WKH/DGGHU³DJHQHUDOPHWKRGRORJ\IRU HPSLULFDO PRGHOLQJ VKRXOG LQFOXGH DOO WKH GLIIHUHQW ³VWHSV³ RI WKH /DGGHU³DVVSHFLDOFDVHV $ QDWXUDO ZD\ WR IRUP D XQLI\LQJ UHODWLRQVKLS WKDW LQFOXGHV DOO WKH ³VSHFLDOFDVHV´LQWKHODGGHULVE\PXOWLSO\LQJRUDGGLQJWKHIXQGDPHQWDO IXQFWLRQV VSHFLILHG LQ WKHVH FDVHV 7KLV ZRXOG UHTXLUH LQWURGXFWLRQ RI HQRXJK SDUDPHWHUV WKDW PD\ DOORZ H[WUDFWLRQ RI HDFK RI WKHVH VSHFLDO FDVHV LQGLYLGXDOO\ 2EYLRXVO\ DQ DSSURDFK OLNH WKDW YLRODWHV WKH LPSRUWDQW SULQFLSOH RI SDUVLPRQ\ ZKLFK UHTXLUHV WKDW WKH XQGHUO\LQJ PRGHO EH DV VLPSOH DV SRVVLEOH $Q DOWHUQDWLYH DSSURDFK QHHGV WR EH SXUVXHG &RQVLGHUH[SUHVVLQJIE\ I1(η, ε1; θ1 ) = H[S{(α/λ)[(η+ε1)λ ±1]} = H[S{(α/λ)[(η+σε1Ζ1)λ ± 1]},
(7.5)
ZLWKε1 DVGHILQHGHDUOLHUDQG θ1 ^αλ, σε1} LVDYHFWRURIUHDOYDOXHG SDUDPHWHUV WKDW QHHG WR EH GHWHUPLQHG :H DVVXPH WKDW WKH OLQHDU
&+$37(57+(50002'(/
SUHGLFWRU η ZLWKLWVDGGLWLYHHUURULVVWULFWO\SRVLWLYHVRWKDWλFDQWDNH ERWK QHJDWLYH DQG SRVLWLYH YDOXHV ZLWKRXW FKDQJLQJ WKH PRQRWRQH LQFUHDVLQJ FKDUDFWHU RI WKH UHODWLRQVKLS )RU DQ\ QRQSRVLWLYH /3 D FKDQJHRIORFDWLRQFDQDFKLHYHWKHUHTXLUHGSURSHUW\ ,WLVHDV\WRUHDOL]HWKDWE\DSURSHUVHOHFWLRQRISDUDPHWHUV YDOXHVDOO RIWKHVL[ FDVHV LQFOXGHGLQ WKH ODGGHU PD\ EH GHULYHG IURP 7KLV ZLOOEHGHPRQVWUDWHGVKRUWO\ 7KH5000RGHO &RPELQLQJ DQG WKHJHQHUDOPRGHOIRUWKHUHODWLRQVKLS EHWZHHQWKHUHVSRQVHDQGWKHYDULRXVYDULDWLRQJHQHUDWLQJDJHQWV GHQRWH LWE\I LV < = I (η, ε1, ε2; θ) = I1(η, ε1; θ1) I2(ε2; θ2) = H[S{(α/λ)[(η+ε1)λ − 1] + µ2 + ε2} = H[S{(α/λ)[(η+σε1Ζ1)λ − 1] + µ2 + σε2Ζ2},
(7.6)
RU : = ORJ(<) = (α/λ)[(η+ε1)λ − 1] + µ2 + ε2 = (α/λ)[(η+σε1Ζ1)λ − 1] + µ2 + σε2Ζ2,
(7.7)
ZKHUH = DQG = GHULYH IURP D ELYDULDWH VWDQGDUG QRUPDO GLVWULEXWLRQ ZLWK FRUUHODWLRQ ρ DQG θ ^α, λ, µ ρ σε1, σε2} LV D YHFWRU RI SDUDPHWHUVWKDWQHHGWREHGHWHUPLQHG. 7R PRWLYDWH WKH H[SUHVVLRQ LQ WKUHH H[DPSOHV RI FRPPRQO\ HQFRXQWHUHGFRQYH[IXQFWLRQVZLOOEHDGGUHVVHG)RUVLPSOLFLW\DVVXPH WKDW WKHUH LV QR UDQGRP YDULDWLRQ ε1=ε2=) 7KHQ DV λ O WKH H[SUHVVLRQ (1/λ)[(η)λ ± 1] O ORJ η 7KLVLPSOLHVWKDWIRUα=1, λ=ZH REWDLQDOLQHDUUHODWLRQVKLS< η. )RUα≠DQGλ ZHREWDLQDSRZHU UHODWLRQVKLS< ηα)RUλ ≠ DQH[SRQHQWLDOUHODWLRQVKLSLVREWDLQHG 7KH UHDGHU PD\ OLNHZLVH HDVLO\ VKRZ WKDW $QWRLQH HTXDWLRQ DQ H[SRQHQWLDOSRZHUUHODWLRQVKLSUHIHUWR&KDSWHU LVDOVRDVSHFLDOFDVH RI ,Q IDFW QXPHURXV VFLHQWLILF DQG HQJLQHHULQJ PRGHOV ZLWK PRQRWRQH FRQYH[ UHODWLRQVKLSV FDQ EH VKRZQ WR EH VSHFLDO FDVHV RI
5(63216(02'(/,1*0(7+2'2/2*<
7KLVLVWKHVXEMHFWRI&KDSWHUZKHUHZHGHPRQVWUDWHIRUVRPH RIWKHPRGHOVLQ&KDSWHUWKDWWKH\DUHLQGHHGVSHFLDOFDVHVRIWKH500 PRGHO $OOWKHFDVHVLQFOXGHGLQWKH/DGGHUDUHQRZVKRZQWREHVSHFLDO FDVHVRI 7RREWDLQ&DVH OLQHDUUHODWLRQVKLS 6HWλ α 7RREWDLQ&DVH SRZHUUHODWLRQVKLS 6HW λ α ≠ 7RREWDLQ&DVH H[SRQHQWLDO 6HW λ 7RREWDLQ&DVH H[SRQHQWLDOSRZHU /HWλ ≠λ ≠ 7RREWDLQ&DVHV H[SRQHQWLDOH[SRQHQWLDO DQG H[SRQHQWLDO H[SRQHQWLDOSRZHU 5HLQWURGXFH WKH OLQHDU SUHGLFWRU ZLWK WKH HUURU LQ WKH IRUP RI I QDPHO\ LQWURGXFH IRU ησε1= LQ H[S^(βN >(ησε1= N@` $SSDUHQWO\WKHYDOXHVβ N UHLQWURGXFHWKHOLQHDUSUHGLFWRUDV LQ +RZHYHUN≠LQWURGXFHVDQH[SRQHQWLDOH[SRQHQWLDOSRZHU UHODWLRQVKLS &DVHVZLWKFRQYH[LW\LQWHQVLW\OLNHWKDWRI&DVHVDQGDUHUDUHLQ SUDFWLFH :H ZLOO WKHUHIRUH DVVXPH KHQFHIRUWK WKDW WKH DERYH YDOXHV RI β=1 DQGN DUHWKHDFWXDOYDOXHVLQWKHJHQHUDO500PRGHO 1RWHKRZHYHUWKDWWKHLQWURGXFWLRQRIWZRDGGLWLRQDOSDUDPHWHUVLI UHLWHUDWHG PD\ LQ IDFW DOORZ XV WR DFKLHYH LQFUHDVLQJ GHJUHHV RI FRQYH[LW\ WR D GHVLUDEOH GHJUHH 7KXV WKH EDVLF SDWWHUQ RI ³OLQHDU SRZHU H[SRQHQWLDO´ FDQ EH UHSHDWHG DW ZLOO ZLWK WKH LQWURGXFWLRQ RI DGGLWLRQDO SDUDPHWHUV DV ZH KDYH MXVW GHPRQVWUDWHG :LWK I PRGHOHG E\ WKH ³/DGGHU´ LV UHQGHUHG IURP D GLVFUHWH KLHUDUFK\ RI IXQGDPHQWDOFRQYH[IXQFWLRQVLQWRDFRQWLQXRXVFRQYH\RU FRQWLQXRXV FRQYH[LW\ LI \RX ZLOO WKDW PD\ FDUU\ WKH PRGHOHU LQ D FRQWLQXRXV IDVKLRQIURPRQHVWHSRIWKHODGGHUWRWKHQH[WZKLOHWKHODGGHULWVHOILV XQERXQGHGIURPDERYH-DFRE VODGGHUFRPHVWRPLQG &RQVLGHU QH[W WKH GXDOHUURU VWUXFWXUH RI WKH PRGHO $V LQGLFDWHG HDUOLHU LW LV DVVXPHG WKDW = DQG = RULJLQDWH LQ D ELYDULDWH VWDQGDUG QRUPDOGLVWULEXWLRQZLWKFRUUHODWLRQρ7KHFRQGLWLRQDOGLVWULEXWLRQRI= JLYHQ= ]LVQRUPDOZLWKPHDQρ]DQGYDULDQFHρ:HPD\ZULWH
&+$37(57+(50002'(/
IRU=LQ Ζ2 | ]1 = ρ]1 + (1−ρ2)(1/2) Ζ,
(7.8)
ZKHUH=LVDVWDQGDUGQRUPDOYDULDEOHLQGHSHQGHQWRI=,QWURGXFLQJIRU =IURP LQWR DQG ZHREWDLQUHVSHFWLYHO\ < H[S{(α/λ)[(η+σε1Ζ1)λ − 1] + µ2 + σε2[ρΖ1+(1−ρ2)(1/2) Ζ2]}, (7.9) : ORJ(<) = (α/λ)[(η+σε1Ζ1)λ − 1] + µ2 + σε2[ρΖ1+(1−ρ2)(1/2) Ζ2], (7.10) ZKHUH IRU XQLIRUPLW\ LQ QRWDWLRQ ZH UHSODFHG = ZLWK = QDPHO\ = LV QRZ DVVXPHG WR EH D VWDQGDUG QRUPDO YDULDEOH ZKLFK LV XQFRUUHODWHG LQGHSHQGHQW RI= (TXLYDOHQWO\E\FRQGLWLRQLQJ=RQ= ]ZHREWDLQ : ORJ < җ = (α/λ){[η + σε1[ρΖ2+(1−ρ2)(1/2) Ζ1]]λ − 1} + µ2 + σε2Ζ2. (7.11) (TXDWLRQV DQG GLVSOD\ WKH 500 PRGHO LQ LWV PRVW JHQHUDO IRUP %RWK PRGHOV H[SUHVV WKH UHVSRQVH LQ WHUPV RI WZR XQFRUUHODWHG VWDQGDUG QRUPDO YDULDEOHV DQG WKH WZR PRGHOV DUH HTXLYDOHQWLQWKHVHQVHWKDWRQHPRGHOFDQEHGHULYHGIURPWKHRWKHU 6RPH YDULDWLRQV RI WKH EDVLF PRGHO DV UHSUHVHQWHG E\ RU PD\EHGHYHORSHGE\SXUVXLQJGLIIHUHQWGHILQLWLRQVIRUWKHHUURUV WKLV ZDV EULHIO\ DOOXGHG WR HDUOLHU DQG UHLQWURGXFHG KHUH LQ D PRUH FRQVLVWHQWIDVKLRQ &RQVLGHUWKHHUURUVWUXFWXUHLQ DQG )RUVPDOOHUURUε2 << ZHPD\ZULWH 1 + ε2 ≈ H[S(ε2). )RUVPDOOHUURU ε1 << ηZHPD\ZULWH (η+ε1)λ = ηλ(1+ε1/η)λ ≈ ηλH[S(λε1/η). 7KHVH WZR SRVVLEOH YHUVLRQV IRU HDFK HUURU WHUP JLYH ULVH WR IRXU GLIIHUHQW HUURUVWUXFWXUHV IRU WKH EDVLF PRGHO &RXSOHG ZLWK WKH WZR HTXLYDOHQW YDULDWLRQV ZKHUH WKH FRUUHODWHG HUURUWHUPV KDYH EHHQ UH H[SUHVVHGLQWHUPVRIWZRLQGHSHQGHQWVWDQGDUGQRUPDOYDULDEOHV> DQG @ WKH 500 PRGHO PD\ EH GLVSOD\HG LQ GLIIHUHQW IRUPV 7KHVH DUH LQWURGXFHG EHORZ 2QO\ YDULDWLRQV RI WKH HUURUVWUXFWXUHV DUH
5(63216(02'(/,1*0(7+2'2/2*<
VKRZQ7KHUHDGHUPD\LQWURGXFHIRUHDFKYDULDWLRQWKHHUURUVLQWHUPVRI WKHQRQFRUUHODWHGVWDQGDUGQRUPDOYDULDWHVVLPLODUO\WRWKHGHULYDWLRQRI DQG 0RGHO7KHEDVLFPRGHO < H[S{(α/λ)[(η+ε1)λ − 1] + µ2 + ε2}. 0RGHO'LIIHUHQWSUHVHQWDWLRQRIε2 < H[S{(α/λ)[(η+ε1)λ − 1] + µ2 }(1+ε2) .
(7.6D)
0RGHO'LIIHUHQWSUHVHQWDWLRQRIε1 < H[S {(α/λ)[ηλ H[S(λε1 / η) − 1] + µ2 + ε2}. 0RGHO'LIIHUHQWSUHVHQWDWLRQRIERWKε1DQGε2 < H[S {(α/λ)[ηλ H[S (λε1 / η) − 1] + µ2 }(1+ε2).
(7.6E) (7.6F)
)RUHDFKPRGHOZHFDQLQWURGXFH (LWKHUIRUεε2: ε2 = σε2[ρΖ1+(1−ρ2)(1/2) Ζ2],
(7.12)
ZKHUHε1 = σε1=DQG=DQG=DUHXQFRUUHODWHG RUIRUεε1: ε1= σε1[ρΖ2 + (1−ρ2)(1/2) Ζ1],
(7.13)
ZKHUHε2 = σε2=DQG=DQG=DUHXQFRUUHODWHG
7KHVH GLIIHUHQW IRUPXODWLRQV RI WKH HUURUV LQWURGXFH GHVLUDEOH IOH[LELOLW\ LQWR WKH VWUXFWXUH RI WKH TXDQWLOHIXQFWLRQ RI WKH 500 HUURU GLVWULEXWLRQDVHODERUDWHGRQLQ&KDSWHU 7KH5HVSRQVH0RPHQWV ([DFW PRPHQWV RI WKH UHVSRQVH DV JLYHQ E\ RU PD\ EH REWDLQHG E\ QXPHULFDO LQWHJUDWLRQ ZLWK UHVSHFW WR = DQG = 6LQFH WKH
&+$37(57+(50002'(/
ODWWHU DUH LQGHSHQGHQW ZH PD\ VHSDUDWH LQWHJUDWLRQ ZLWK UHVSHFW WR = IURP WKDW ZLWK UHVSHFW WR = )RU WKH UWK QRQFHQWUDO PRPHQW RI < PRPHQWDERXW]HUR ZHREWDLQ µU = Ε(
U
T
H[S{(Uα/λ)[(η+σε1]1)λ−1] + Uσε2ρ]1 } φ(]1)G]1}
[{
T
U
U
U
H[S [Uµ2 + Uσε2(1−ρ2)(1/2) ]2] φ(]2)G]2},
(7.14)
ZKHUHφ LVWKHVWDQGDUGQRUPDOGHQVLW\IXQFWLRQ GI $OWHUQDWLYHO\ IURP DQ DSSUR[LPDWH KLJKO\ DFFXUDWH GHQVLW\ IXQFWLRQ IRU : GHYHORSHGLQ &KDSWHU DQ DSSUR[LPDWH GI IRU < LV GHYHORSHG ZKHUHRIDOOPRPHQWVRI<PD\EHHDVLO\HYDOXDWHGQXPHULFDOO\ $SSUR[LPDWHH[SOLFLWH[SUHVVLRQVIRUWKHPRPHQWVRIWKHUHVSRQVH< PD\ EH REWDLQHG ZLWK WKH KHOS RI D 7D\ORU VHULHV H[SDQVLRQ 7R GHULYH WKHVHPRPHQWVZULWH DVDSURGXFWRIWZRIXQFWLRQVRQHLQWHUPVRI =DQGWKHRWKHULQWHUPVRI= < H[S{(α/λ)[(η+σε1=1) λ − 1] + σε2ρΖ1} [H[S {µ2 + σε2(1−ρ2)(1/2) Ζ2} = J1(Ζ1;ω1) J2(Ζ2; ω2 ),
(7.15)
ZKHUH ω ^α, λ ρ, σε σε2` ω ^µ2,σε2,ρ` DQG J DQG J DUH H[SUHVVLRQVWKDWUHODWHWR=DQGWR=UHVSHFWLYHO\ 6LQFH ERWK H[SUHVVLRQV DUH H[SRQHQWLDO IXQFWLRQV RI WKH UHVSHFWLYH QRUPDO YDULDEOHV DQG EHFDXVH RIWKH VWDWLVWLFDO LQGHSHQGHQFH RI = DQG = WKH UWK QRQFHQWUDO PRPHQW RI < PD\ EH GHULYHG DV WKH SURGXFWRI WKH PHDQV RI J DQG RI J ZKHUH IRU WKH PHDQ RI J ZH UHSODFH ^α σε2} E\ ^Uα, Uσε2} DQG VLPLODUO\ IRU WKH PHDQ RI J ZH UHSODFH ^µ σε2`E\^UµUσε2`7KHUWKQRQFHQWUDOPRPHQWRI<LV Ε(
(7.16)
7KHIXQFWLRQ J LV D ORJQRUPDO YDULDEOH )RU WKH UWK PRPHQW RI J ZHKDYH H[DFW
5(63216(02'(/,1*0(7+2'2/2*<
Ε{[J2(Ζ2;ω2)]U} = H[S[Uµ2 + (1−ρ2)(Uσε2)2/2].
(7.17)
5HIHUIRUH[DPSOHWR6WXDUWDQG2UG 6HFWLRQ ([SDQGLQJ J LQWR D 7D\ORU VHULHV LQ WHUPV RI SRZHUV RI = DURXQG ]HURDQGWDNLQJH[SHFWDWLRQRIWKHILUVWIRXUWHUPVZHREWDLQIRUWKHUWK QRQFHQWUDOPRPHQWRIJ Ε{[J1(Ζ1;θ1)]U} ≅ H[S[(Uα/λ)(ηλ − 1)] [ {1 + (1/2)[(Uα)σε12(λ−1)ηλ − 2 + U2(ασε1ηλ−1 + ρσε2)2]}.
(7.18)
)URP DQG DQ H[SOLFLW DSSUR[LPDWH H[SUHVVLRQ IRU WKH UWKQRQFHQWUDOPRPHQWRI<LV Ε(
(7.19))
)URP WKH QRQFHQWUDO PRPHQWV µU (
)RUWKHQRUPDOGLVWULEXWLRQ6N .X )RUWKHPHDQDQGWKHYDULDQFHZHREWDLQDSSUR[LPDWHO\ µ(<) = H[S[(α/λ)(ηλ − 1) + µ2 + (1−ρ2)(σε2)2/2)] [{1 + (1/2)[ασε12(λ−1)ηλ − 2 + (ασε1ηλ−1 + ρσε2)2]},
(7.20)
&+$37(57+(50002'(/
9DU < = H[S[(2α/λ)(ηλ − 1) + 2µ2 + (1−ρ2)(2σε2)2/2)] [{1+(1/2)[(2α)σε12(λ−1)ηλ − 2 + 4(ασε1ηλ−1+ρσε2)2]} − [µ(<)]2, µ(:) = Ε[ORJ < ] = (α/λ)[(η)λ −1] + (1/2)α(λ−1)σε12ηλ−2 + µ2,
(7.21)
ZKHUHµ : LVDJDLQGHYHORSHGXVLQJD7D\ORUH[SDQVLRQDSSUR[LPDWLRQ $VVXPHPRPHQWDULO\WKDWσε1= σε2=([DPLQLQJWKHH[SUHVVLRQVIRU WKHPHDQVLQ RQHPD\UHDOL]HKRZYHUVDWLOH500LVLQPRGHOLQJ YDULRXVO\VKDSHGOLQNIXQFWLRQV DOLQNIXQFWLRQLVDWUDQVIRUPDWLRQRIWKH UHVSRQVH PHDQ QHHGHG WR REWDLQ WKH OLQHDU SUHGLFWRU ,Q SDUWLFXODU WKH 500 PHDQ SURYLGHV D ODUJH PDMRULW\ RI WKH OLQN IXQFWLRQV XVHG LQ WKH IUDPHZRUN RI */0 7KXV IRU λ DQG α ZH REWDLQ D OLQHDU OLQN IXQFWLRQZKLOHλ DQGα UHVXOWLQDUHFLSURFDOUHODWLRQVKLSEHWZHHQ WKHOLQHDUSUHGLFWRUDQGWKHPHDQW\SLFDOWRWKHJDPPDHUURUGLVWULEXWLRQ $ YDOXH RI λ UHVXOWV LQ D ORJ OLQN IXQFWLRQ W\SLFDO WR WKH 3RLVVRQ GLVWULEXWLRQ 2QH UHDOL]HV WKDW WKH YDOXHV RI WKH SDUDPHWHUV LQ SDUWLFXODU WKDW RI λ, DUHFULWLFDOLQGHWHUPLQLQJWKHILQDOVWUXFWXUHRIWKHPRGHO6LQFHλLV HVWLPDWHG ZLWKLQ WKH GDWDDQDO\VLV SURFHGXUH UHIHU WR WKH HVWLPDWLRQ SURFHGXUHVGHYHORSHGLQ&KDSWHU WKHILQDOIRUPRIWKHPRGHOZRXOG EHGHWHUPLQHGVROHO\E\WKHGDWDRQKDQG1RVSHFLILFDWLRQRIWKHPRGHO VWUXFWXUH SULRU WR GDWD DQDO\VLV LV QHHGHG LQ FRPSOLDQFH ZLWK 5HTXLUHPHQW &KDSWHU &KDSWHULQWURGXFHVHVWLPDWLRQSURFHGXUHVIRUWKHSDUDPHWHUVRIWKH 500PRGHO ([SORULQJWKH5HODWLRQVKLSEHWZHHQWKH&9DQGη η )RUUHODWLRQDOPRGHOVLWLVDFRPPRQO\HQFRXQWHUHGSKHQRPHQRQWKDWDV WKH OLQHDU SUHGLFWRU LQFUHDVHV WKH YDULDQFH WHQGV WR VWDELOL]H DQG LWV UHODWLYHYDOXH UHODWLYHWRWKHPHDQ WHQGVWRGHFUHDVH,QIDFWWKHZHOO NQRZQSURSHUW\RIWKHGHFRXSOLQJRIWKHYDULDQFHIURPWKHPHDQDVWKH PHDQ EHFRPHV ODUJH KDV DOUHDG\ EHHQ DOOXGHG WR LQ 5HTXLUHPHQW &KDSWHU
5(63216(02'(/,1*0(7+2'2/2*<
,WLVWKHUHIRUHRILQWHUHVWWRH[DPLQHZKHWKHUWKH500PRGHORZQV WKLV FRPPRQO\ HQFRXQWHUHG IHDWXUH ,Q 6HFWLRQ ZH SURYLGH DQ LQWXLWLYHH[SODQDWLRQRIZK\WKH500DOORZVGHFRXSOLQJRIWKHYDULDQFH IURP WKH PHDQ ZKHQ WKH OLQHDU SUHGLFWRU EHFRPHV ODUJH :H EDVH WKLV LQWXLWLYH H[SODQDWLRQ RQ WKH IDFW WKDW WKH HUURU WHUP DVVRFLDWHG ZLWK WKH OLQHDU SUHGLFWRU ε1, EHFRPHV QHJOLJLEOH DQG WKHUHIRUH WKH YDULDQFH RI ORJ < WHQGVWRVWDELOL]H$PRUHULJRURXVPDWKHPDWLFDOSURRIZLOOQRZ EHGHYHORSHG &RQVLGHUWKHYHUVLRQRIWKH500PRGHOJLYHQE\ E 5HZULWLQJLW LQ WHUPV RI LQGHSHQGHQW VWDQGDUG QRUPDO YDULDEOHV = DQG = DVVXPH ρ ZHKDYH < H[S{(α/λ)[ηλ H[S (λσε1Ζ1 / η) − 1] + µ2 + σε2Ζ2}. (7.22) 5HODWHWRWKHORJQRUPDOYDULDEOHLQVTXDUHGEUDFNHWV7KLVLVWKHRQO\ UDQGRP YDULDEOH LQ KDYLQJ SDUDPHWHUV WKDW GHSHQG RQ WKH OLQHDU SUHGLFWRU :H ZLOO ILQG RXW KRZ LWV FRHIILFLHQW RI YDULDWLRQ &9 σ/µ FKDQJHV ZLWK η. 'HQRWLQJ WKH ORJQRUPDO YDULDEOH E\ 8 ZH KDYH IURP 8 H[S[λ ORJ(η) + (λσε1/η)Ζ1] = H[S[µ1 + σ1Ζ1].
(7.23)
)RU D ORJQRUPDO YDULDEOH 8 ZLWK SDUDPHWHUV µ1 DQG σ1, WKH PHDQ DQG WKH YDULDQFH DUH UHIHU WR -RKQVRQ .RW] DQG %DODNULVKQDQ &KDSWHURUWR6WXDUWDQG2UG 6HFWLRQ ( 8 H[S(µ1 + σ12/2), 9DU 8 H[S(2µ1 + σ12)[H[S (σ12)−1],
(7.24)
DQGIRUWKHVTXDUHGFRHIILFLHQWRIYDULDWLRQZHREWDLQ (&9)2 = 9DU 8 )/[Ε(8)]2 = H[S(2µ1 + σ12)[H[S(σ12)−1] /H[S(2µ1 + σ12) = H[S(σ12)−1.
(7.25)
,QWURGXFLQJIRUσ12 LQWHUPVRIWKH500SDUDPHWHUV σ12 (λσε1/η) >UHIHUWR @ZHREWDLQRQGLIIHUHQWLDWLQJ ZLWKUHVSHFWWRWKH OLQHDUSUHGLFWRUη ∂(&9)2 / ∂η = − (2/η)(λσε1/η)2 H[S[(λσε1/η)2]. (7.26)
&+$37(57+(50002'(/
$VVXPLQJ WKDW WKH OLQHDU SUHGLFWRU η LV SRVLWLYH LW LV REYLRXV WKDW &9LVDGHFUHDVLQJIXQFWLRQRIWKHOLQHDUSUHGLFWRU)XUWKHUPRUHDVWKH OLQHDU SUHGLFWRU WHQGV WR LQILQLW\ &9 WHQGV WR ]HUR QDPHO\ WRZDUGV D VWDEOHYDULDQFH 5HIHUHQFHV >@ -RKQVRQ 1 / .RW] 6 %DODNULVKQDQ 1 &RQWLQXRXV 8QLYDULDWH 'LVWULEXWLRQV²9ROXPHQG(GLWLRQ-RKQ:LOH\ 6RQV >@6KRUH+ 0RGHOLQJDUHVSRQVHZLWKVHOIJHQHUDWHGDQGH[WHUQDOO\JHQHUDWHG VRXUFHVRIYDULDWLRQ4XDOLW\(QJLQHHULQJ >@ 6WXDUW $ 2UG - . .HQGDOO¶V $GYDQFHG 7KHRU\ RI 6WDWLVWLFV 9 'LVWULEXWLRQ7KHRU\&KDUOHV*ULIILQ &RPSDQ\/RQGRQ
This page intentionally left blank
&KDSWHU
(VWLPDWLQJWKH5HODWLRQDO0RGHO
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
5(63216(02'(/,1*0(7+2'2/2*<
&KDSWHU DQG D FHUWDLQ YDULDWLRQ RI WKH EDVLF PRGHO 7KHVH DUH UHVSHFWLYHO\ < H[S^ αλ > ησεң= λ±@µσε2[ρ= ρ =@` < H[S^ α/λ)[ηλH[S λσε1=η ±@µ2 + σε2[ρ= ρ =@` (TXDWLRQ LVDYDULDWLRQRI ZKLFKPD\EHPRUHFRQYHQLHQW WRDSSO\ZKHQVHDUFKURXWLQHVDUHHPSOR\HGWRLGHQWLI\0/HVWLPDWHVIRU WKH SDUDPHWHUV 0RUH VSHFLILFDOO\ WKH WHUP ησε1= λ LQ PD\ RFFDVLRQDOO\GHOLYHULPDJLQDU\YDOXHVZKHQDQHVWLPDWHRIλLVVHDUFKHG 7KLVSUREOHPLVHOLPLQDWHGZKHQPRGHO LVXVHG 7KHHVWLPDWLRQSURFHGXUHVGHYHORSHGKHUHPD\EHHDVLO\H[WHQGHGWR WKHRWKHUYDULDWLRQVRIWKH500PRGHO /HWXVVSHFLI\WKHWKUHHVHWVRISDUDPHWHUV ,3DUDPHWHUVDVVRFLDWHGZLWKWKH/3 η ββ;βN;N
,,3DUDPHWHUVDVVRFLDWHGZLWKWKHQRQOLQHDUVWUXFWXUHRIWKHPRGHO ^α, λ, µ2`
,,,3DUDPHWHUVDVVRFLDWHGZLWKWKHHUURUV ^ρ, σε1, σε2}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
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
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
5(63216(02'(/,1*0(7+2'2/2*<
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λ $OWKRXJK SURFHGXUHV KDYH EHHQ GHYLVHG WR HVWLPDWH FRQFXUUHQWO\ERWKWKHFRHIILFLHQWVRIWKH/3DQGWKHSDUDPHWHU V RIWKH UHVSRQVH WUDQVIRUPDWLRQ LQ PRVW VWDWLVWLFDO SDFNDJHV FXUUHQWO\ RQ WKH PDUNHWWKHWZRVWDJHVRIPRGHOLQJDUHFDUULHGRXWVHTXHQWLDOO\ 6LPLODUO\ LQ PRGHOLQJ YLD */0 ZH GHWHUPLQH LQ DGYDQFH WKH VWUXFWXUHRIWKHILQDO SRVVLEO\QRQOLQHDU PRGHOE\ DVVXPLQJDFHUWDLQ OLQN IXQFWLRQ D WUDQVIRUPDWLRQ RI WKH UHVSRQVH PHDQ QHHGHG WR REWDLQ
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
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«LWLVLQWHUHVWLQJWKDWWKHFRQFOXVLRQVUHJDUGLQJ WKH UROHV RI WKH YDULDEOHV DUH VRPHZKDW GLIIHUHQW VXJJHVWLQJ WKDW WKH DQDO\VWPD\ZLVKWRWU\PRUHWKDQRQHOLQN ,QTXDOLW\HQJLQHHULQJWKHLPSOLFDWLRQVRIWKHODWWHUREVHUYDWLRQPD\ EH SDUWLFXODUO\ ERWKHUVRPH :KLOH LQ VFLHQWLILF PRGHOLQJ DQG DW WLPHV DOVR LQ HQJLQHHULQJ PRGHOLQJ ERWK WKH VWUXFWXUH RI WKH /3 DQG WKH
5(63216(02'(/,1*0(7+2'2/2*<
VWUXFWXUHRIWKHILQDO SRVVLEO\QRQOLQHDU PRGHOPD\EHRISUDFWLFDORU WKHRUHWLFDO LQWHUHVW LQ TXDOLW\ HQJLQHHULQJ RIWHQWLPHV WKH PRGHOHU LV LQWHUHVWHGH[FOXVLYHO\LQLGHQWLI\LQJWKHHIIHFWVWKDWUHODWHVLJQLILFDQWO\WR WKHUHVSRQVH QDPHO\LQWKHVWUXFWXUHRIWKH/3 6XFK D VFHQDULR LV HQFRXQWHUHG IRU H[DPSOH LI WKH H[SHULPHQW DQG WKH SXUVXLQJ DQDO\VHV DUH LQWHQGHG RQO\ DV GLDJQRVWLF WRROV LQ GHWHU PLQLQJVLJQLILFDQWHIIHFWVIROORZLQJZKLFKVRPHFRUUHFWLYHDFWLRQFDQEH WDNHQ,QWKLVFDVHWKHH[DFWVWUXFWXUHRIWKHUHODWLRQVKLSEHWZHHQWKH/3 DQGWKHUHVSRQVHPD\EHRIVHFRQGDU\LPSRUWDQFHRUQRWLPSRUWDQWDWDOO
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
6WDJH,$SSUR[LPDWLQJDWUDQVIRUPHGUHVSRQVHYLDD7D\ORU VHULHVH[SDQVLRQDQGHVWLPDWLQJWKHSDUDPHWHUVYLD&&$ /HW < EH D UHVSRQVH DVVXPHG WR EH H[SUHVVHG LQ VWDQGDUGL]HG XQLWV VR WKDWLWKDV]HURPHDQDQGXQLWYDULDQFH/HW^;;;N`EHDVHWRI. SUHGLFWRUV IDFWRUV RU SDUDPHWHUIUHH IXQFWLRQV RI IDFWRUV WKDW PD\ SRWHQWLDOO\ WUDQVPLW VWDWLVWLFDOO\ VLJQLILFDQW YDULDWLRQ WR < /HW η GHQRWH WKH /3 RI D VWUXFWXUH \HW XQNQRZQ DQG DVVXPH WKDW D SUHOLPLQDU\ H[DPLQDWLRQRIWKHGDWDVXJJHVWVWKDWWKHUHODWLRQVKLSEHWZHHQ<DQGη GRHVQRWMXVWLI\OLQHDUUHJUHVVLRQDQDO\VLV WKHUHODWLRQVKLSLVQRQOLQHDU WKH DVVXPSWLRQ RI D FRQVWDQW HUURU YDULDQFH LV YLRODWHG E\ WKH GDWD RU PRUH JHQHUDOO\ QR NQRZQ QRUPDOL]LQJ WUDQVIRUPDWLRQ FDQ UHYRNH WKH QRUPDOVFHQDULR /HW * < GHQRWH DQ XQVSHFLILHG UHODWLRQVKLS EHWZHHQ WKH UHVSRQVH DQGWKH/3QDPHO\* < η+εZKHUHεLVWKHHUURU1RWHWKDW* < LV D UHVSRQVHWUDQVIRUPDWLRQUHTXLUHG WR DFKLHYH DGGLWLYLW\ RIHIIHFWV LWLV QRWDWUDQVIRUPDWLRQRIWKHPHDQOLNHLQ*/0 'HYHORSLQJ * < LQWR D 7D\ORU VHULHV H[SDQVLRQ DURXQG ]HUR WKH UHVSRQVHPHDQ DQGNHHSLQJWKHILUVWSWHUPVZHREWDLQ * < ≅* * <* <*S <SS ZKHUH *M LV WKH MWK GHULYDWLYH FDOFXODWHG DURXQG < DQG S LV WKH QXPEHU RI WHUPV LQ WKH 7D\ORU H[SDQVLRQ DSDUW IURP WKH FRQVWDQW FRQVLGHUDWLRQVLQGHWHUPLQLQJSDUHGLVFXVVHGODWHURQ 'HILQH <7M <MM M S WRUHH[SUHVV* < DV
* < ≅%%<7%S<7S
ZLWKFRHIILFLHQWV^%M`WKDWQHHGWREHHVWLPDWHGIURPWKHGDWD /HWWKH/3EHDVLQ :HZLVKWRGHWHUPLQHWKHSDUDPHWHUV^%M` M S VRWKDW* < UHSUHVHQWHGE\WKH5+6RI PD\VHUYHDV WKHUHVSRQVHLQDVWHSZLVHOLQHDUUHJUHVVLRQSURFHGXUHWKDWZLOOGHWHUPLQH WKH FRHIILFLHQWV VLJQLILFDQFH DQG HVWLPDWH RI WKH /3 ,Q RWKHU ZRUGVRXUPRGHOLV ^7UDQVIRUPHGUHVSRQVHD7D\ORUVHULHVH[SDQVLRQ` /3HUURU
5(63216(02'(/,1*0(7+2'2/2*<
RU
%%<7%S<7S η + ε ββ;βN;N ε
7KHHVWLPDWLQJSURFHGXUHFRPSULVHVWZRVWDJHV ,Q 6WDJH , &&$ LV LPSOHPHQWHG WR GHWHUPLQH FRHIILFLHQWV ^β L` L . RI DQ LQLWLDO /3 DQG WKH ILQDO FRHIILFLHQWV RI WKH ILWWHG 7D\ORU DSSUR[LPDWLRQ ^%M` M S 7KLV VWDJH RI WKH DQDO\VLV LV H[SHFWHGWRGHOLYHUFDQRQLFDOYDULDEOHVWKDWPD[LPDOO\FRUUHODWH* < DV DSSUR[LPDWHG E\ ZLWK η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` )RU H[DPSOH LI H[SDQVLRQ WR
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
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
5(63216(02'(/,1*0(7+2'2/2*<
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
([DPSOH7D\ORUH[SDQVLRQXSWRWHUP <7 <7 <7 ([DPSOH7D\ORUH[SDQVLRQXSWRWHUP <7 <7 <7
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
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
5(63216(02'(/,1*0(7+2'2/2*<
% :KLFKHIIHFWVWRLQFOXGHLQWKHLQLWLDO/3 6WDJH, " 7KLV LV LQGHHG VWLOO DQ RSHQ TXHVWLRQ ZKLFK VHHPV WR UHTXLUH IXUWKHU H[SORUDWLRQ EHIRUH D FRQFOXVLYH DQVZHU FDQ EH JLYHQ 7KHUH DUH WZR RSWLRQV WR SXUVXH 2QH PD\ ZLVK WR LQFOXGH DOO SRVVLEO\ DIIHFWLQJ SUHGLFWRUV HYHQWKRVHWKDWPD\ODWHUWXUQRXWWREHLQVLJQLILFDQW RUHOVH LQFOXGH LQ WKH LQLWLDO /3 RQO\ PDLQ HIIHFWV DJDLQ LUUHVSHFWLYH RI ZKLFK DUH VLJQLILFDQW $V GHPRQVWUDWHG LQ WKH WZR H[DPSOHV RI 6HFWLRQ LQFOXVLRQ LQ WKH LQLWLDO /3 RI 6WDJH , RQO\ PDLQ HIIHFWV GRHV QRW EDU RWKHU HIIHFWV OLNH LQWHUDFWLRQV IURP HQWHULQJ DV VLJQLILFDQW SUHGLFWRUV LQWR WKH ILQDO /3 LQ 6WDJH ,, :H PD\ ZLVK WR FRQVLGHU WKH RSWLRQ RI LQFOXGLQJRQO\PDLQHIIHFWVLQWKHLQLWLDO/3LIWKHQXPEHURISRWHQWLDOO\ VLJQLILFDQW PDLQ HIIHFWVLVODUJH ,Q WKDW FDVHLQFOXVLRQ RI VHFRQG RUGHU WHUPV RUKLJKHU PD\WXUQWKHLPSOHPHQWDWLRQRI&&$WRRFXPEHUVRPH GXHWRWKHODUJHVHWRIYDULDEOHVWKDWFRPSULVHVWKHLQLWLDO/3 &RQYHUVHO\ DOO SRWHQWLDOO\ VLJQLILFDQW HIIHFWV SUHGLFWRUV PD\ EH LQFOXGHGLQWKHLQLWLDO/3NQRZLQJWKDWLIDFHUWDLQHIIHFWLVXQUHODWHGWR WKHUHVSRQVHLWVZHLJKWLQWKHFDQRQLFDOYDULDEOHWKDWUHSUHVHQWVWKHLQLWLDO /3ZLOOEHVPDOO&RQVHTXHQWO\WKDWHIIHFWZLOOKDYHQHJOLJLEOHLQIOXHQFH LQ GHWHUPLQLQJ WKH FRHIILFLHQWV RI WKH ILWWHG 7D\ORU DSSUR[LPDWLRQ IRU * < 0RUHULJRURXVJXLGHOLQHVUHJDUGLQJWKHFRPSRVLWLRQRIWKHLQLWLDO/3 KDYH \HW WR EH GHYHORSHG &XPXODWLYH HPSLULFDO H[SHULHQFH ZRXOG SUREDEO\ EH KHOSIXO LQ GHYHORSLQJ WKHVH JXLGHOLQHV ,Q WKH QXPHULFDO H[DPSOHVRI6HFWLRQDQG&KDSWHURQO\PDLQHIIHFWVDUHLQFOXGHG LQWKHLQLWLDO/3RI6WDJH, & :KDWDUHWKHDVVXPSWLRQVDQGDUHWKH\YDOLG" ,QFRUSRUDWLQJDILWWHG7D\ORUDSSUR[LPDWLRQDVWKHUHVSRQVHLQDOLQHDU UHJUHVVLRQPRGHOUHTXLUHVWKDWWKHDVVXPSWLRQVXQGHUO\LQJWKLVPRGHOEH VSHFLILHGDQGVKRZQWREHYDOLG,QSDUWLFXODURQHKDVWRVKRZWKDWWKH DVVXPSWLRQ UHODWLQJ WR WKH KRPRJHQHLW\ RI WKH UHVLGXDO YDULDQFH DQ HVVHQWLDO DVVXPSWLRQ WR WKH LPSOHPHQWDWLRQ RI XQZHLJKWHG OLQHDU UHJUHVVLRQLVYDOLG6LQFHWKHFDQRQLFDOYDULDEOHXVHGDVWKHUHVSRQVHLQ WKHOLQHDUUHJUHVVLRQSDUWRIWKHDQDO\VLVLVD7D\ORUVHULHVDSSUR[LPDWLRQ
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
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εZHKDYH
(>ORJ <ε @ (α/λ)[(η+ε1)λ ±@µ
* < η + ε1 ^>ORJ < µ2](λ/α) + 1}1/λ
RU
ZKHUHLQ ORJ < HVWLPDWHV(>ORJ <ε @ ,QDFFRUGDQFHZLWKWKHDVVXPSWLRQVRIWKH500PRGHO* < LQ LV QRUPDO ZLWK PHDQ η DQG FRQVWDQW YDULDQFH σε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
5(63216(02'(/,1*0(7+2'2/2*<
$ VRPHZKDW FRPSDUDEOH DSSURDFK ZDV SXUVXHG E\ /LQ DQG 9RQHVK LQ GHWHUPLQLQJ WKH SDUDPHWHU λ RI WKH %& WUDQVIRUPDWLRQ 7KH\ KDYH FRQVWUXFWHG D QRQOLQHDU UHJUHVVLRQ PRGHO ZKLFK LV XVHG WR HVWLPDWH WKH SDUDPHWHU λ LQ VXFK D IDVKLRQ WKDW WKH QRUPDO SUREDELOLW\ SORW RI WKH GDWD RQ WKH WUDQVIRUPHG VFDOH LV DV FORVH WR OLQHDULW\ DV SRVVLEOH $QDORJRXVO\ WKH QHZ DSSURDFK LGHQWLILHV YLD &&$ D 7D\ORU DSSUR[LPDWLRQIRUDWUDQVIRUPDWLRQRI<WKDWEHVWFRUUHODWHVZLWKDOLQHDU FRPELQDWLRQRIWKHFDQGLGDWHHIIHFWV7KLVLVWDQWDPRXQWWRWKHFKRLFHRI WKH EHVW UHVSRQVH WUDQVIRUPDWLRQ WKDW FDQ WKHQ EH XVHG WR FRQVWUXFW D YDOLGOLQHDUUHJUHVVLRQPRGHO )LQDOO\QRWHWKDWVLQFHOLQHDUUHJUHVVLRQDQDO\VLVLVXVHGLQ6WDJH,, GLDJQRVWLFFKHFNLQJ IRUWKH YDOLGLW\ RI WKH QRUPDO VFHQDULR PD\ DOZD\V EHFRQGXFWHG*LYHQWKHIRUHJRLQJGLVFXVVLRQLWLVKLJKO\OLNHO\WKDWWKH GLDJQRVWLF FKHFNLQJ ZRXOG MXVWLI\ LPSOHPHQWDWLRQ RI OLQHDU UHJUHVVLRQ DQDO\VLV ' ,VWKHUHDVLQJOHFRUUHFW/3" 7KHQHZDSSURDFKLPSOLFLWO\DVVXPHVWKDWWKHUHLVRQO\DVLQJOHFRUUHFW /3 DQG WKDW WKLV /3 PD\ EH LGHQWLILHG LUUHVSHFWLYH RI WKH QRQOLQHDU PRGHOHPSOR\HGWRUHSUHVHQWWKHUHODWLRQVKLSRIWKH/3ZLWKWKHUHVSRQVH EHLWD%&WUDQVIRUPDWLRQDOLQNIXQFWLRQRUDILWWHG500PRGHO *LYHQWKHFRXSOLQJEHWZHHQWKHWZRSKDVHVRIWKHHVWLPDWLRQSURFHVV ZKLFK H[LVWV ZLWK PRVW FXUUHQW DSSURDFKHV RQH PD\ UHDVRQDEO\ DUJXH WKDWWKH/3LVHVVHQWLDOO\GHSHQGHQWRQWKHQRQOLQHDUPRGHOVHOHFWHGDQG WKDWWKHWZRPRGHOLQJFRPSRQHQWVFDQQRWEHVHSDUDWHG GHFRXSOHG 7KLV GHSHQGHQF\ ZDV HDUOLHU GHPRQVWUDWHG E\ D TXRWH IURP 0\HUV HW DO ZKHUH D ORJ OLQN DQG D VTXDUHURRW OLQN SURGXFHG GLIIHUHQW /3V GLIIHUHQW HIIHFWV LQFOXGHG HYHQ WKRXJK JRRGQHVVRIILW VWDWLVWLFV ZHUH WKHVDPH*LYHQWKHODWWHUFULWHULRQRQHPD\DGYRFDWHWKDWGLIIHUHQW/3V PD\EHDGRSWHGDVHTXDOO\YDOLG :HEHOLHYHRWKHUZLVH,IFHUWDLQHIIHFWVDQGQRWRWKHUVWXUQRXWWREH VLJQLILFDQWO\UHODWHGWRWKHUHVSRQVHWKHQWKHVHHIIHFWVUHIOHFWUHDOLW\DQG WKH\VKRXOGEHDGGUHVVHGDQGQRWRWKHUVLQPDNLQJGHFLVLRQVUHJDUGLQJ WKH UHVSRQVH ZKHWKHU WKHVH GHFLVLRQV UHODWH WR SUHGLFWLRQ FRQWURO RU TXDOLW\LPSURYHPHQWV 0D[LPL]LQJWKHFRUUHODWLRQEHWZHHQWKH/3DQG
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
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
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
SORWVLQRUGHUWRGHWHFWRXWOLHUV7KLVRSWLRQLVDYDLODEOHLQDQ\VRIWZDUH SURFHGXUHWKDWFRQGXFWV&&$ 3KDVH(VWLPDWLQJWKH5000RGHO ,QWURGXFWLRQ 6DPSOHYDOXHV RI WKH /3 REWDLQHG LQ 3KDVH PD\ VHUYH DV LQSXW WR HVWLPDWHDVLQJOHUHJUHVVRU SUHGLFWRU 500PRGHOLQ3KDVH &RQVLGHU WKH ORFDWLRQ RI WKH /3 )URP ZH UHDOL]H WKDW WR FDOFXODWHWKHPHDQWKH/3QHHGVWREHUDLVHGWRWKHSRZHURIλ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ηLZL`LVDYDLODEOHZKHUH ZLLVWKHORJWUDQVIRUPHGUHVSRQVHYDOXHIRUREVHUYDWLRQLDQGVLPLODUO\ ηLLVWKHYDOXHRIWKH/36LQFHWKHODWWHUKDVDOUHDG\EHHQHVWLPDWHGLQ 3KDVHRIWKHDQDO\VLVZHQHHGWRHVWLPDWHWZRVHWVRISDUDPHWHUV DV GHWDLOHGLQWKHLQWURGXFWRU\6HFWLRQ 6HW,,3DUDPHWHUVRIWKHQRQOLQHDUVWUXFWXUHRIWKHPRGHO {α, λ, µ2` 6HW,,,3DUDPHWHUVDVVRFLDWHGZLWKWKHHUURUWHUPV^ρ, σε1, σε2}. 7KHHVWLPDWLRQSURFHGXUHRI3KDVHLVDFFRUGLQJO\GLYLGHGLQWRWZR VWDJHV
5(63216(02'(/,1*0(7+2'2/2*<
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α, λ, µ2 ` $ZHLJKWHGQRQOLQHDUOHDVWVTXDUHV :1//6 ZRXOGPLQLPL]H
6
Q
>Z L
L
L : @ >! L : @
ZKHUH µL : DQG σL : DUH WKH PHDQ DQG WKH VWDQGDUG GHYLDWLRQ RI REVHUYDWLRQ ZL L Q 0\HUV HW DO TXRWH WKH ZRUN RI :HGGHUEXUQ ZKLFK VXJJHVWV WKDW XVH RI ZHLJKWHG OHDVWVTXDUHV SURGXFHV DV\PSWRWLF SURSHUWLHV WKDW DUH TXLWH VLPLODU WR WKRVH RI PD[LPXP OLNHOLKRRG 0/ HVWLPDWRUV 7KXV LI JRRG HVWLPDWHV RI WKH YDULDQFHV RI LQGLYLGXDO REVHUYDWLRQV DUH DYDLODEOH PLQLPL]LQJ ZRXOG SURYLGH QHDUO\ 0/ HVWLPDWHV IRU WKH SDUDPHWHUV RI WKH 500 PRGHO 1RWHWKDWVLQFHWKH/3KDVDOUHDG\EHHQLQGHSHQGHQWO\HVWLPDWHGRQO\ WKH 500 SDUDPHWHUV ^α, λ, µ2}, DUH HVWLPDWHG DW WKLV VWDJH RI WKH DQDO\VLV 9DULRXV GLDJQRVWLFV FKHFNLQJ URXWLQHO\ DYDLODEOH LQ DQ\ VWDWLVWLFDO SDFNDJH WKDW LQFOXGH D PRGXOH IRU 1//6 PD\ EH XVHG WR MXGJH WKH TXDOLW\ RI WKH GHULYHG HVWLPDWHV )RU H[DPSOH FRUUHODWLRQV EHWZHHQ HVWLPDWHV RU HVWLPDWHV FRQILGHQFH LQWHUYDOV PD\ EH GHULYHG 7KLVZLOOEHGHPRQVWUDWHGLQWKHQXPHULFDOH[DPSOHVRI6HFWLRQ /HW XV GHYHORS H[SUHVVLRQV IRU WKH PHDQ DQG WKH YDULDQFH LQ )RUWKHEDVLFPRGHODQGLWVYDULDWLRQ> DQG UHVSHFWLYHO\@
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
( : (^ αλ > ησε1= λ±@`µ2
( : α/λ ^H[S>λORJ η λσε1/η @`µ2
ZKHUHLQ WKHH[SUHVVLRQLQVTXDUHGEUDFNHWVLVWKHPHDQRIDORJ QRUPDOO\ GLVWULEXWHG UDQGRP YDULDEOH 1RWH WKDW ERWK DQG LPSO\WKDWWKHPHDQGRHVQRWLQFOXGHWKHHUURUSDUDPHWHUV^σε2, ρ` 8VLQJ D 7D\ORU H[SDQVLRQ IRU WKH H[SUHVVLRQ LQ VTXDUHG EUDFNHWV LQ DQG WDNLQJ H[SHFWDWLRQ ZLWK UHVSHFW WR = ZH PD\ XVH WKH H[SUHVVLRQJLYHQLQ WRREWDLQ
( : ≅ αλ > η λ±@ α λ σε1ηλµ
D
7KH /3 η) KDV DOUHDG\ EHHQ HVWLPDWHG DQG WKHUHIRUH RQO\ WKH IRXU SDUDPHWHUV ^α, λ, µ2, σε1} QHHG WR EH HVWLPDWHG E\ PLQLPL]DWLRQ RI :H SUHIHU WR HVWLPDWH DOO SDUDPHWHUV DVVRFLDWHG ZLWK WKH HUURUV ^ρ, σε1, σε2` E\ D VHSDUDWH HVWLPDWLRQ URXWLQH WKDW ZLOO EH H[SRXQGHG VKRUWO\ 7KLV LPSOLHV WKDW LQ PLQLPL]LQJ RQO\ WKH 500 SDUDPHWHUV ^α, λ, µ2` ZLOO EH HVWLPDWHG 7KH SDUDPHWHU σε1 ZLOO EH HVWLPDWHGLQWKHSXUVXLQJVWDJHWRJHWKHUZLWKWKHRWKHUHUURUSDUDPHWHUV UHIHUWR6XEVHFWLRQ $QLQLWLDOHVWLPDWHWRXVHLQ IRUWKH ILUVWLWHUDWLRQRQO\ PD\EHWDNHQIURPWKHUHVLGXDOVWDQGDUGGHYLDWLRQRI WKHOLQHDUUHJUHVVLRQ DQDO\VLV RI 3KDVH $OWHUQDWLYHO\ RQH PD\ RSW WR LQFOXGH σε1 DV DQ DGGLWLRQDO SDUDPHWHU WR EH HVWLPDWHG YLD WKH 1//6 SURFHGXUH 7KH ILUVW LWHUDWLRQ LQ WKH HVWLPDWLRQ SURFHGXUH GLIIHUV UHODWLYH WR VXFFHVVLYH LWHUDWLRQV DOVR LQ WKDW QR HVWLPDWHV RI ^σL : ` DUH \HW DYDLODEOH 7KLV ZLOO EH WUXH XQOHVV UHSHDW REVHUYDWLRQV UHSOLFDWHV H[LVW VRWKDWSXUHHUURUHVWLPDWHVRIYDULDQFHDUHDYDLODEOH,QDOORWKHUFDVHV XQZHLJKWHG1//6LVFRQGXFWHGIRUWKHILUVWLWHUDWLRQ ,QSXUVXLQJLWHUDWLRQVXSGDWHGHVWLPDWHVRI^σL : `ZLOOEHDYDLODEOH VRWKDWZHLJKWHG1//6FDQEHXVHG $SSURSULDWHGHWHUPLQDWLRQRI WKH ZHLJKWVLQ LV FUXFLDOIRU WKH HIIHFWLYHHVWLPDWLQJRIWKH500VWUXFWXUHSDUDPHWHUV7KHZHLJKWVDUH UHFLSURFDO YDOXHV RI WKH YDULDQFH RI : IRU LQGLYLGXDO REVHUYDWLRQV 7R GHULYHWKHVHZHLJKWVDVVXPHWKDWDOOSDUDPHWHUVRIWKH500PRGHODUH NQRZQ RUWKDWXSGDWHGHVWLPDWHVIRUWKHVHSDUDPHWHUVKDYHEHHQGHULYHG DWWKHFRPSOHWLRQRI6WDJH,,RIWKHDQDO\VLV (VWLPDWHVRIYDULDQFHVIRU
136
RESPONSE MODELING METHODOLOGY
individual observations may then be derived numerically either from the quantile function [(8.1) or (8.2)] or from the density function of W. Both methods will now be developed for (8.1). Corresponding expressions may similarly be developed for (8.2). Estimating the Variance of W (Based on Eq. 8.1) Method I Re-write (8.1) as W = Q1(Z1; ω1) + Q2(Z2; ω2),
(8.13)
where Q1(Z1; ω1) = (α/λ)[(η+σε1Z1)λ – 1] + µ2 + σε2ρZ1, Q2(Z2; ω2) = σε2(1-ρ2)(1/2) Z2,
(8.14)
with, respectively, parameters' vectors ω1 = {α, λ, µ2, σε1, σε2ρ}, ω2 = σε2(1-ρ2)(1/2).
(8.15)
Note, that ω1 contains five parameters while ω2 has only one parameter. Since Z1 and Z2 are independent random variables so are Q1 and Q2, and for the r-th non-central moment of W we obtain E(Wr) = µr'(W) =
r
r
j =0
∑ j M
j
(Q1 ) M r − j (Q2 ) ,
(8.16)
where Mj(Q1) and Mr-j(Q2) are the j-th and (r-j)-th non-central moments of Q1 and Q2, respectively, namely: Mk(Q1) =
∫
∞
−∞
{(α/λ)[(η+σε1z)λ – 1] + µ2 + σε2ρz}k φ(z)dz (8.17)
Mk(Q2) =
∫
∞
−∞
[σε2(1-ρ2)(1/2) z]k φ(z)dz
= [σε2(1-ρ2)(1/2)]k µk(Z) where µk(Z) is the k-th standard normal moment.
(8.18)
CHAPTER 8 ESTIMATING THE RELATIONAL MODEL
137
The variance of W for individual observations may be estimated from (8.16)-(8.18), and inserted into the minimization routine (8.10). Method II This method uses an approximate expression for the density function of W, fW, given in (8.20) below. The variance of W may then be estimated from µr'(W) =
∫
∞
−∞
w r f W ( w)dw ,
Var(W) = µ2'(W) - [µ1'(W)]2 ,
(8.19)
where µr'(W) is the r-th non-central moment of W. Note, that applying Method I may encounter difficulties if the expression (η+σε1Z1) becomes negative during numerical integration (imaginary values may be encountered, especially if η has values close to zero). Therefore, we may either select a translation constant for η that ensures non-negative values for (η+σε1Z1) (we have discussed earlier why such a translation is legitimate), or we may opt for using Method II in calculating weights for the W-NL-LS procedure. 8.4.3. Stage II - Estimating the RMM "Error Parameters" {ρ, σε1, σε2} At this stage, the error parameters, {ρ, σε1, σε2}, are estimated via a ML routine, based on the approximate log-likelihood function of the RMM error distribution. In Section 9.1, an expression for the density function (d.f.) of W is developed, which requires numerical integration. This may render application of an ML estimation procedure a numerically prohibitive undertaking. We therefore opt for a Taylor expansion which would eliminate the need for integration. The result is a relatively simple expression for the d.f. of W, which is based on the expected values of the first four terms in the Taylor expansion of the conditional density fWε1(wε1,θ ). The resulting expression appears in (9.10), where η=1. With η as an additional argument, we have:
5(63216(02'(/,1*0(7+2'2/2*<
U
I: Z θ T I:ε Zε1 θ θ Iε1 ε Gε U
≅>π(1−ρ2)σε22]−1/2 H[S> =@ [^ σε12[(1−ρ2)1/2σε2]−2[(αηλ−1+ρσε2/σε1)2(= αηλ−2 λ ρ2)1/2σε2=@`
ZKHUH θ θ ^αλ, µ2, ρ, σε1, σε2}, DQG = ^Z αλ > η λ±@µ`[ ρ2)σε22]1/2.
1RWHWKDWWKLVLVDQDSSUR[LPDWHH[SUHVVLRQIRUWKHGIRI:ZKHQ WKH 500 PRGHO LV JLYHQE\ :KHQWKH 500PRGHO LV WKH DSSUR[LPDWH H[SUHVVLRQ IRU I: LV WKH VDPH H[FHSW IRU WKH ODVW WHUP LQ ZKHUHWKHWHUP λ−1 LVUHSODFHGZLWKλ. 7KHUHVXOWVGHYHORSHGKHUHPD\HDVLO\EHDGDSWHGIRURWKHUYDULDWLRQV RI WKH 500 PRGHO DV JLYHQ LQ &KDSWHU +HQFHIRUWK ZH ZLOO UHODWH RQO\WRPRGHO ZLWKWKHGIJLYHQE\ )URP DQG WKHORJOLNHOLKRRGIXQFWLRQLV // θ θ Z
Q L
U
ORJ T I:ε ZLε1 θ θ Iε1 ε Gε U
≅ Q ORJ>π ρ2)σε22] +
Q L
Q L
> =L@
ORJ^ σε12[(1−ρ2)1/2σε2]−2
[ [(αηLλ−1+ρσε2/σε1)2(=L αηLλ−2 λ ρ2)1/2σε2=L@`
ZKHUHIRUWKHLWKREVHUYDWLRQ
=L >ZL αλ > ηL λ±@µ@[ ρ2)σε22]1/2
0D[LPL]DWLRQRI ZLWKUHVSHFWWRWKHHUURUSDUDPHWHUVSURYLGHV 0/HVWLPDWHVIRU{ρ, σε1, σε2} 6XPPDU\RIWKHHVWLPDWLRQSURFHGXUH 3KDVH $VVXPHWKDWWKHYDOXHVRIWKH/3IRULQGLYLGXDOREVHUYDWLRQVDUHNQRZQ HVWLPDWHG IURP3KDVHRIWKHDQDO\VLV , 0LQLPL]H 6 WR REWDLQ LQLWLDO HVWLPDWHV IRU ^α, λ, µ2` 8VH XQ
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
ZHLJKWHG1//6IRUWKLVVWHS$VDQHVWLPDWHIRUσε12XVHDQHVWLPDWHRI WKH UHVLGXDO YDULDQFH REWDLQHG IURP WKH VWHSZLVH OLQHDU UHJUHVVLRQ RI 3KDVH$OWHUQDWLYHO\ILQGHVWLPDWHVIRUDOOSDUDPHWHUV^α, λ, µ2, σε1` ,,)RUWKHGHULYHGHVWLPDWHVRI^α, λ, µ2`PD[LPL]HWKHORJOLNHOLKRRG IXQFWLRQ WRILQGHVWLPDWHVIRUWKHHUURUV SDUDPHWHUV^ρ, σε1, σε2}. ,,,&DOFXODWHYDULDQFHVIRULQGLYLGXDOREVHUYDWLRQVXVLQJHLWKHU0HWKRG , RU 0HWKRG ,, 8VH SXUH HUURU YDULDQFHV LI DYDLODEOH 8VH UHFLSURFDO YDOXHV RI WKH GHULYHG YDULDQFHV DV ZHLJKWV IRU WKH QH[W LWHUDWLRQ 6WHS ,9 ,9 'HULYH XSGDWHG HVWLPDWHV IRU ^α, λ, µ2`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
5(63216(02'(/,1*0(7+2'2/2*<
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
5XQ
)DFWRUV
2EVHUYDWLRQV
$
%
&
'
(
)
*
\
\
\
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
7DEOH([DPSOH0DLQ&&$5HVXOWV Canonical Correlation 1 2 3
0.792078 0.567448 0.202167
Adjusted Canonical Correlation
Approximate Standard Error
Squared Canonical Correlation
0.750096 0.501351 0.010847
0.054939 0.099966 0.141416
0.627387 0.321997 0.040871
Eigenvalue Differ. Propor. Cumulat. Ratio
1 2 3
1.6838 0.4749 0.0426
1.2088 0.4323
0.7649 0.2157 0.0194
0.7649 0.9806 1.0000
0.2423 0.6503 0.9591
FNum. Den. Value DF DF Pr > F 3.25 1.52 0.33
21 12 5
106 76 39
<.0001 0.1354 0.8903
UHIHUWR6KRUHSIRUMXVWLILFDWLRQ 7KHVHGDWDDUHQRZUHDQDO\]HGLPSOHPHQWLQJWKHQHZDSSURDFK )LUVWWKHUHVSRQVHLVVWDQGDUGL]HG PHDQ 67' DQGWKUHH QHZYDULDEOHVDUHGHILQHG^<7M` M IRUFKRRVLQJ. UHIHUWR DQ HDUOLHU GLVFXVVLRQ LQ 6HFWLRQ $SSO\LQJ &&$ XVLQJ D 6$6 SURFHGXUH 7DEOH LV REWDLQHG 1RWH WKDW RQO\ PDLQ HIIHFWV ZHUH LQFOXGHGLQWKH/3IRUWKH&&$ 2QO\ WKH ILUVW FDQRQLFDO FRUUHODWLRQ LV VLJQLILFDQWO\ ODUJHU WKDQ ]HUR S )XUWKHUPRUH WKH ILUVW HLJHQYDOXH DFFRXQWV IRU RIWKHYDULDWLRQLQWKHGDWD7KHFRUUHVSRQGLQJHTXDWLRQIRUWKH ILUVWFDQRQLFDOYDULDEOH LQWHUPVRIWKHUDZFRHIILFLHQWV LV : <7<7±<7 6WDQGDUGL]LQJ : WR KDYH ]HUR PHDQ DGGLQJ DQG XQLW YDULDQFH LW PD\ QRZ EHFRPH WKH UHVSRQVH IRU D PXOWLSOH VWHSZLVH UHJUHVVLRQDQDO\VLVWKHUHVXOWVRIZKLFKDUHGLVSOD\HGLQ7DEOH7KH IROORZLQJ PRGHO LV REWDLQHG ZKLFK LQFOXGHV DOO HVWLPDWHVVLJQLILFDQW DW WKHOHYHO UHIHUWR7DEOH : $%&( *
$ & ± % '
ZLWK5 DGM DQGPRGHO)UDWLRRI S 7KHURRWPHDQ VTXDUHG HUURU LV /DFNRIILW WHVW SURYHV WR EH KLJKO\ QRQ VLJQLILFDQW S
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH([DPSOH0DLQ5HVXOWVIURP/LQHDU5HJUHVVLRQ$QDO\VLV5HVSRQVHLVWKH 6WDQGDUGL]HG&DQRQLFDO9DULDWH:
6XPPDU\RI)LW 56TXDUH 56TXDUH$GM 5RRW0HDQ6TXDUH(UURU 0HDQRI5HVSRQVH 2EVHUYDWLRQV RU6XP:JWV
H
$QDO\VLVRI9DULDQFH
6RXUFH 0RGHO (UURU &7RWDO
')
6XPRI6TXDUHV
0HDQ6TXDUH
)5DWLR 3URE!)
/DFN2I)LW
6RXUFH /DFN2I)LW 3XUH(UURU 7RWDO(UURU
')
6XPRI6TXDUHV
0HDQ6TXDUH
)5DWLR 3URE!) 0D[56T
3DUDPHWHU(VWLPDWHV 7HUP ,QWHUFHSW $ % & ( * $ & % '
(VWLPDWH
6WG(UURU
W5DWLR 3URE!_W_
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
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σε1 GHULYHG IURP WKH OLQHDU UHJUHVVLRQ DQDO\VLV RI 3KDVH QDPHO\ σε1= WKHLQLWLDOHVWLPDWHVREWDLQHGDUH α = λ µ 1RWH WKDW WKH H[SUHVVLRQ IRU WKH PHDQ ( : HPSOR\HG LQ WKH 1//6 SURFHGXUHGRHVQRWLQFOXGHHLWKHUρRUσε2 7KHVH HVWLPDWHV DUH LQWURGXFHG LQWR DQG WKH ODWWHU LV PD[LPL]HGZLWKUHVSHFWWRρ, σε1DQGσε2WRREWDLQ ρ σε1 = σε2 // 7KH XSGDWHG YDOXH RI σε1 GRHV QRW DOWHU WKH YDOXHV RI WKH 500 SDUDPHWHUVJLYHQHDUOLHUDQGWKHUHIRUHQRIXUWKHULWHUDWLRQVZHUHQHHGHG &DOFXODWLQJ YDULDQFHV IRU LQGLYLGXDO REVHUYDWLRQV XVLQJ 0HWKRG ,, VKRZVWKHYDULDQFHσ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
5(63216(02'(/,1*0(7+2'2/2*<
SDUDPHWHUV HVWLPDWHVZHUHREWDLQHG α ^`λ ^` µ2 = ^`.
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` DQG WKH RWKHU VHW LQFOXGLQJ ^<7<7` RQ WKH UHDVRQV IRU VHOHFWLQJD7D\ORUH[SDQVLRQRIDVHFRQGGHJUHHUHIHUWR6HFWLRQDQG 7DEOH UHVXOWVLQ7DEOH7KHUHLVDVLQJOHVLJQLILFDQWHLJHQYDOXH DVVRFLDWHG ZLWK D FDQRQLFDO FRUUHODWLRQ RI S 7KH ILUVW FDQRQLFDO YDULDEOH DVVRFLDWHG ZLWK WKH VHWRI YDULDEOHV GHULYHG IURP WKH 7D\ORUH[SDQGHGUHVSRQVHWUDQVIRUPDWLRQLV
: <7 <7
6WDQGDUGL]LQJ : WR KDYH ]HUR PHDQ DQG XQLW YDULDQFH LW PD\ QRZ EHFRPH WKH UHVSRQVH IRU D PXOWLSOH VWHSZLVH UHJUHVVLRQ WR \LHOG WKH IROORZLQJPRGHO
: ;±; ; ±;;±;;±;;ε
ZLWK5 DGM DQGPRGHO)UDWLRRI :LWKDQG GHJUHHVRI IUHHGRP IRU WKH PRGHO DQG WKH HUURU UHVSHFWLYHO\ WKH ODWWHU UHVXOW LV KLJKO\VLJQLILFDQW S $OOPDLQHIIHFWVDUHKLJKO\VLJQLILFDQWWRR S 7KHURRWPHDQVTXDUHHUURULV 1RWH WKDW DV ZLWK WKH ILUVW H[DPSOH DOWKRXJK RQO\ PDLQ HIIHFWV SDUWLFLSDWHG LQ WKH LQLWLDO /3 RI 6WDJH , VLJQLILFDQW LQWHUDFWLRQV ZHUH
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
7DEOH([DPSOH([SHULPHQWDO'HVLJQDQG2EVHUYDWLRQV [
[
[
[
\
7DEOH([DPSOH0DLQ&&$5HVXOWV Canonical Correlation Analysis
Canonical Correlation 1 2
0.904635 0.330063
Adjusted Canonical Correlation
Approximate Standard Error
Squared Canonical Correlation
0.882968 0.164909
0.046898 0.230070
0.818365 0.108942
Eigenvalue Differ. Propor. Cumulat. Ratio
1 2
4.5056 0.1223
4.3833
0.9736 0.0264
0.9736 1.0000
0.1618 0.8911
FNum. Den. Value DF DF Pr > F 3.71 0.45
8 3
20 11
0.0082 0.7235
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH([DPSOH&RPSDULVRQRIREVHUYHGYDOXHVDQG500SUHGLFWHGYDOXHV
2EV
6DPSOH \
500
2EV
6DPSOH \
500
LQFOXGHGLQWKHILQDO/3GHULYHGLQ6WDJH,,RIWKHDQDO\VLV )RU3KDVHRIWKHDQDO\VLVDYDOXHRILVILUVWDGGHGWRWKH/3WR HQVXUHWKDWDOO/3YDOXHVREWDLQHGLQ3KDVHDUHQRQQHJDWLYH$ILUVW UXQ RI 1//6 ZLWK : DV WKH UHVSRQVH DQG DQ LQLWLDO HVWLPDWH RI σε1 JLYHVLQLWLDOHVWLPDWHVRI α = λ µ2 = 7KH ORJ WUDQVIRUPDWLRQ λ= LV DJDLQ WKH ULJKW WUDQVIRUPDWLRQ WR DSSO\WRWKHUHVSRQVH 0D[LPL]LQJ WR GHULYH HVWLPDWHV IRU ^σε1, σε2, ρ} ZH UHDOL]H WKDW IRU ρ> QRQDGPLVVLEOH QHJDWLYH HVWLPDWHV IRU σε1 DUH REWDLQHG. )RU ρ 0/ HVWLPDWHV RI ERWK σε1 DQG σε2 DUH QRQ QHJDWLYH KRZHYHU WKH YDOXH RI WKH ORJOLNHOLKRRG IXQFWLRQ KDUGO\ FKDQJHVDQGVWD\VDWWKHFRQVWDQWYDOXHRIDERXW// 7KHUHIRUH DQ DUELWUDU\ YDOXH LV VHOHFWHG RI ρ WR REWDLQ σε1= σε2= . ,QVHUWLQJ WKH XSGDWHG YDOXH RI σε1 LQWR WKH HVWLPDWLRQ SURFHGXUH IRU WKH 500 SDUDPHWHUV ^α, λ, µ2}, GRHV QRW DOWHU WKHLU HVWLPDWHVDQGWKHUHIRUHQRIXUWKHULWHUDWLRQVZHUHUHTXLUHG 7DEOHGLVSOD\VSUHGLFWHGUHVSRQVHYDOXHVQH[WWRDFWXDO REVHUYHG YDOXHV ,Q WHUPV RI JRRGQHVVRIILW WKH UHVXOWV VXJJHVW DFFHSWDEOH JRRGQHVVRIILW)XUWKHUFRPSDULVRQVUHODWHGWRRWKHU PDMRUDSSURDFKHV %&WUDQVIRUPDWLRQDQG*/0 ZLOOEHJLYHQLQ&KDSWHU
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
6RPH IXUWKHU UHOHYDQW UHIHUHQFHV GHDOLQJ ZLWK WKH HVWLPDWLRQ SURFHGXUHVPD\EHIRXQGLQ6KRUH DEFG 5HIHUHQFHV >@&RQGUD/: 5HOLDELOLW\,PSURYHPHQWZLWK'HVLJQRI([SHULPHQWV0DUFHO 'HNNHU1< >@+DPDGD01HOGHU-$ *HQHUDOL]HGOLQHDUPRGHOVIRUTXDOLW\LPSURYHPHQW H[SHULPHQWV-RXUQDORI4XDOLW\7HFKQRORJ\ >@ /LQ / , 9RQHVK ( ) $Q HPSLULFDO QRQOLQHDU GDWDILWWLQJ DSSURDFK IRU WUDQVIRUPLQJGDWDWRQRUPDOLW\$PHULFDQ6WDWLVWLFLDQ >@ 0\HUV 5 + 0RQWJRPHU\ ' & $ WXWRULDO RQ JHQHUDOL]HG OLQHDU PRGHOV -RXUQDORI4XDOLW\7HFKQRORJ\ >@0\HUV5+0RQWJRPHU\'&9LQLQJ** *HQHUDOL]HG/LQHDU0RGHOV ZLWK$SSOLFDWLRQVLQ(QJLQHHULQJDQGWKH6FLHQFHV-RKQ:LOH\ 6RQV >@ 6KRUH + 0RGHOLQJ D QRQQRUPDO UHVSRQVH IRU TXDOLW\ LPSURYHPHQW ,QWHUQDWLRQDO-RXUQDORI3URGXFWLRQ5HVHDUFK >@6KRUH+ 0RGHOLQJDUHVSRQVHZLWKVHOIJHQHUDWHGDQGH[WHUQDOO\JHQHUDWHG VRXUFHVRIYDULDWLRQ4XDOLW\(QJLQHHULQJ >@ 6KRUH + D 5HVSRQVH 0RGHOLQJ 0HWKRGRORJ\ 500 &XUUHQW VWDWLVWLFDO GLVWULEXWLRQV WUDQVIRUPDWLRQV DQG DSSUR[LPDWLRQV DV VSHFLDO FDVHV RI 500 &RPPXQLFDWLRQVLQ6WDWLVWLFV 7KHRU\DQG0HWKRGV > @ 6KRUH + E 5HVSRQVH 0RGHOLQJ 0HWKRGRORJ\ 500 9DOLGDWLQJ HYLGHQFH IURP HQJLQHHULQJ DQG WKH VFLHQFHV 4XDOLW\ DQG 5HOLDELOLW\ (QJLQHHULQJ ,QWHUQDWLRQDO >@6KRUH+ F 7KH5DQGRP)DWLJXH/LPLW0RGHODVDVSHFLDOFDVHRIWKH500 PRGHO D FRPPHQW RQ3DVFXDO D &RPPXQLFDWLRQVLQ6WDWLVWLFV 6LPXODWLRQ DQG&RPSXWDWLRQ >@6KRUH+ G 5HVSRQVH0RGHOLQJ0HWKRGRORJ\ 500 0D[LPXPOLNHOLKRRG HVWLPDWLRQSURFHGXUHV&RPSXWDWLRQDO6WDWLVWLFVDQG'DWD$QDO\VLV,QSUHVV >@ :HGGHUEXUQ 5 : 0 4XDVL/LNHOLKRRG IXQFWLRQV JHQHUDOL]HG OLQHDU PRGHOVDQGWKH*DXVV1HZWRQPHWKRG%LRPHWULFV
$SSHQGL[$&DQRQLFDO&RUUHODWLRQ$QDO\VLV%DFNJURXQG ,QWKLVDSSHQGL[ZHGHOLYHUDEULHIEDFNJURXQGRQFDQRQLFDOFRUUHODWLRQ DQDO\VLVEDVHGODUJHO\RQWKH6$6XVHUJXLGH&DQRQLFDOFRUUHODWLRQLVD WHFKQLTXH IRU DQDO\]LQJ WKH UHODWLRQVKLS EHWZHHQ WZR VHWV RI YDULDEOHV ZKHUH HDFK VHW FDQ FRQWDLQ D GLIIHUHQW QXPEHU RI YDULDEOHV $V VXFK FDQRQLFDOFRUUHODWLRQLVDYDULDWLRQRQWKHFRQFHSWRIPXOWLSOHUHJUHVVLRQ
5(63216(02'(/,1*0(7+2'2/2*<
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
Z E\E\ET\T
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
&+$37(5(67,0$7,1*7+(5(/$7,21$/02'(/
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ρ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ρ WKHQ WKH FRUUHODWLRQ PDWUL[ FDQQRW EH LQYHUWHG DQG &&$ FDQQRW EH SHUIRUPHG ,Q H[WUHPH FDVHV ZKHQWKHDVVXPSWLRQRIODFNRIUHGXQGDQF\LVDOPRVWYLRODWHG QDPHO\
5(63216(02'(/,1*0(7+2'2/2*<
PDQ\ YDULDEOHV DSSURDFK UHGXQGDQF\ WKH VWDWLVWLFDO &&$ SURFHGXUH PD\LVVXHDZDUQLQJHYHQZKHQQRH[DFWUHGXQGDQF\KDVEHHQREWDLQHG ,W LV JRRG SUDFWLFH LQ LPSOHPHQWLQJ &&$ DV LW LV LQ PXOWLSOH OLQHDU UHJUHVVLRQDQDO\VLVWRDYRLGH[WUHPHFDVHRIPXOWLFROOLQHDULW\
&KDSWHU
7KH500(UURU'LVWULEXWLRQ
,QWURGXFWLRQ 9DULDWLRQ LQ WKH OLQHDU SUHGLFWRU /3 SURYLGHV WKH FRPSRQHQW RI V\VWHPDWLF YDULDWLRQ LQ WKH UHVSRQVH WRWDO YDULDWLRQ :KHQ WKH /3 LV FRQVWDQWIOXFWXDWLRQVLQWKHUHVSRQVHDUHUDQGRPDQGWKHLUGLVSHUVLRQLV GHVFULEHGE\WKHHUURUGLVWULEXWLRQ ,Q WKLV FKDSWHU WKH 500 HUURU GLVWULEXWLRQ LV GHULYHG DQG LWV SURSHUWLHVH[SORUHG 6HWWLQJ WKH /3 η WR D FRQVWDQW YDOXH WKH 500 EDVLF PRGHO H[SUHVVHV WKH UHODWLRQVKLS EHWZHHQ WKH UHVSRQVH DQG WKH WZR SRVVLEO\ FRUUHODWHG HUURUVε1 DQGε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
5(63216(02'(/,1*0(7+2'2/2*<
VRPHZHOONQRZQIDPLOLHVRIGLVWULEXWLRQVUHVSRQVHWUDQVIRUPDWLRQVDQG GLVWULEXWLRQDO DSSUR[LPDWLRQV &KDSWHU GHPRQVWUDWHV WKDW WKH 500 HUURUGLVWULEXWLRQPD\LQGHHGVHUYHDVDJRRGSODWIRUPWRPRGHOUDQGRP YDULDWLRQ 7KLV LV GRQH E\ GHULYLQJ DFFXUDWH DSSUR[LPDWLRQV WR VRPH FRPPRQO\DSSOLHGVWDWLVWLFDOGLVWULEXWLRQV 'HULYDWLRQRIWKH500(UURU'LVWULEXWLRQ :LWKRXW ORVV RI JHQHUDOLW\ LQWURGXFH LQ DQG IRU WKH OLQHDU SUHGLFWRU η 7KH UHVSRQVH LV H[SUHVVHG LQ WHUPV RI WKH HUURUV RQ WKH RULJLQDOVFDOH < RULQWKHORJWUDQVIRUPHGVFDOH : E\ < H[S^ αλ > ε1 λ±@µε2` H[S^ αλ > σε1= λ±@µσε2[ρ= ρ =@` : ORJ < αλ > σε1= λ±@µσε2[ρ= ρ =@ ZKHUHε1= σε1=DQGε2= σε2[ρ= ρ =@ DUHWZRHUURUWHUPVIURP D ELYDULDWH QRUPDO GLVWULEXWLRQ ZLWK FRUUHODWLRQ ρ ]HUR PHDQV DQG VWDQGDUGGHYLDWLRQVσε1 DQGσε2, UHVSHFWLYHO\DQG =DQG=DUHWZRQRQ FRUUHODWHG LQGHSHQGHQW VWDQGDUGQRUPDOYDULDEOHV )RU ρ ± UHGXFHV WR WKH RULJLQ ,QYHUVH 1RUPDOL]LQJ 7UDQVIRUPDWLRQ ,17 GHYHORSHGDQGQXPHULFDOO\GHPRQVWUDWHGLQ6KRUH ,QWKDWFDVH µ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
µ :_] θ θ αλ > σε1] λ±@µσε2ρ]
σ(:_] θ θ σε2 ρ
ZKHUHθ θ ^αλ, µ2, ρ, σε1, σε2}. )URPWKLVWKHPDUJLQDO XQFRQGLWLRQDO
&+$37(5 7+(500(5525',675,%87,21
GIRI:LV U
I: Z θ θ T >σ(:_] θ θ ]−1 φ^>Zµ :_] θ θ @>σ(:_] θ θ @`φ ] G] U
DQGφ(.)LVWKHVWDQGDUGQRUPDOGI7KHPDUJLQDO&')RI:LV U
θ) = T Φ^>Zµ :_] θ θ @>σ(:_] θ θ @`φ ])G] ): Z θ U
DQGΦ LVWKHVWDQGDUGQRUPDO&') 7KHGHQVLW\IXQFWLRQRI<GHQRWHLWE\I< \ θ θ LVLQWHUPVRI
I< \ θ θ \ I:>ORJ \ θ θ@
6LPLODUO\WKHGLVWULEXWLRQIXQFWLRQ)< \ θ θ LV
)< \ θ θ ):>ORJ \ θ θ]
)RUλ α σε2 LVWKHQRUPDOGLVWULEXWLRQDQGIRUα ZH REWDLQWKHORJQRUPDOGLVWULEXWLRQ 7KH GLVWULEXWLRQ LV WKH HUURU GLVWULEXWLRQ RI WKH EDVLF 500 PRGHO *LYHQ WKH ZLGH VSHFWUXP RI VFLHQWLILF DQG HQJLQHHULQJ PRGHOV KDYLQJ DQ HUURU GLVWULEXWLRQ GHSLFWHG E\ WKLV PRGHO UHIHU WR &KDSWHU DQG HOVHZKHUH WKLV LV SUREDEO\ RQH RI WKH PRVW FRPPRQ GLVWULEXWLRQV WKDWRQHPD\ILQGLQWKHVFLHQWLILFLQYHVWLJDWLRQRIQDWXUH
θ T I:ε Zε1 θ θ Iε1 ε Gε I: Z θ
U
U
θ @>σ(:ε θ θ @`Iε ε Gε T >σε2 (1−ρ2)1/2 ]−1 φ^>ZLµ :ε θ U
ZKHUHI:ε Zε1 θ θ LVWKHFRQGLWLRQDOGIRI:Iε1 ε LVWKHGIRIWKH QRUPDOHUURUε1, DQG θ αλ > ε1 λ±@µρ(σε2/σε1)ε1 µ :_] θ µ :ε θ θ
σ(:ε θ θ σε2 ρ σ(:_] θ θ
154
RESPONSE MODELING METHODOLOGY
Developing fWε1(wε1; θ), in terms of ε1, into a Taylor series expansion around zero, and then applying the expectation operator to the first four terms (namely, including in the approximation terms up to a third degree), we obtain the relatively simple explicit approximate expression for the d.f of W (η=1) ∞
fW(w;θ) = ∫ fWε1(wε1; θ ) fε1(ε1)dε1 ≅ [2π(1−ρ2)σε22]−1/2 exp[-(1/2)Z2] −∞
x {1+(1/2)σ ε12[(1−ρ2)1/2σε2]−2[(α+ρσε2/σε1)2=2-1) + α(λ−1)(1− ρ2)1/2σε2Z]},
(9.10)
where Z = (w-µ2) / [(1-ρ2)σε22]1/2,
(9.11)
and in taking expectation for the Taylor expansion we introduced E(ε1) = 0, E(ε12 ) = σε12, E(ε13 ) = 0. (9.12) From (9.10), all moments of W may be easily calculated by µ'r(W) = E(Wr)=
∫
∞
−∞
w r f W ( w;θ )dw .
(9.13)
Also, from (9.10), an approximate expression for the d.f. of Y may be derived via (9.6). 9.3. Properties of the Error Distribution The properties of the RMM error distribution are explored. In particular, we investigate the shape versatility by studying the behavior of the RMM error distribution in the (√β1,β2) plane, where √β1 and β2-3 are measures of skewness and kurtosis, respectively [the third and fourth standardized cumulants; Refer to (9.26) below for their definitions]. Recall that in Chapter 6 we have related to adequate coverage of the (√β1,β2) plane as one of the most important requirements for the universal applicability of a methodology for empirical relational modeling (therein, Requirement 7). It is therefore natural that studying the coverage associated with the RMM error distribution is of high priority in order to assess its versatility and that of the RMM relational model.
CHAPTER 9 THE RMM ERROR DISTRIBUTION
155
We start by deriving expressions for the moments of either Y or W (the log-transformed response). If the parameters in (9.1) are specified, then the non-central r-th moment of Y is calculated numerically from ∞
µ'r(Y) = E(Yr) = ∫ (y)r fY(y)dy, −∞
(9.14)
where fY(y) is given by (9.6). However, this method requires double numerical integration, which can be a numerically prohibitive task. Alternatively, use can be made of the fact that Z1 and Z2 in (9.2) are independent to derive either exact or approximate explicit expressions for the moments of W and Y. An exact method was detailed in Chapter 8 for the relational model. It is repeated here with η=1. Re-write (9.2) as W = Q1(Z1; ω1) + Q2(Z2; ω2),
(9.15)
where Q1(Z1; ω1) = (α/λ)[(1+σε1Z1)λ – 1] + µ2 + σε2ρZ1, Q2(Z2; ω2)= σε2(1-ρ2)(1/2) Z2,
(9.16)
with, respectively, parameters' vectors ω = {α, λ, µ2, σε1, σε2ρ}, £
ω2 = σε2(1-ρ2)(1/2).
(9.17)
Note, that ω1 contains five parameters while ω2 has only one parameter. Since Z1 and Z2 are independent random variables so are Q1 and Q2, and for the r-th non-central moment of W we obtain E(Wr) = µr'(W) =
r
r
j =0
∑ j M
j
(Q1 ) M r − j (Q2 ) ,
(9.18)
where Mj(Q1) and Mr-j(Q2) are the non-central j-th and (r-j)-th noncentral moments of Q1 and Q2, respectively: Mk(Q1) =
∫
∞
−∞
{(α/λ)[(1+σε1z)λ – 1] + µ2 + σε2ρz}k φ(z)dz,
Mk(Q2) =
∫
∞
−∞
[σε2(1-ρ2)(1/2) z]k φ(z)dz.
(9.19) (9.20)
From (9.18)-(9.20) all non-central moments of W can be obtained. Note
5(63216(02'(/,1*0(7+2'2/2*<
WKDW FDQEHZULWWHQE\
0N 4 [σε2 ρ @NµN =
D
N
ZKHUHµN = ( = LVWKHVWDQGDUGQRUPDONWKPRPHQW $SSUR[LPDWH H[SOLFLW H[SUHVVLRQV IRU WKH QRQFHQWUDO PRPHQWV RI < FDQEHGHYHORSHGVLPLODUO\WRWKHGHULYDWLRQRIDSSUR[LPDWHH[SUHVVLRQV IRU WKH ILUVW WZR PRPHQWV RI WKH UHVSRQVH LQ WKH UHODWLRQDO PRGHO LQ &KDSWHU 7RGHULYHWKHVHPRPHQWVZULWH DVDSURGXFWRIWZRIXQFWLRQV RQHLQWHUPVRI=DQGWKHRWKHULQWHUPVRI=
< H[S^ αλ > σε1= λ±@ µσε2ρ=`
ω J = ω ω [H[S^σε2 ρ =` J = ω
2 (1/2)
ZKHUH ω1= ^α λ µ2, σε1, σε2ρ` ω ^σε2(1−ρ ) ` DQG J DQG J DUH WZRIXQFWLRQVRI=DQG=UHVSHFWLYHO\ 6LQFH ERWK IXQFWLRQV DUH H[SRQHQWLDO H[SUHVVLRQV RI WKH UHVSHFWLYH QRUPDO YDULDEOHV DQG EHFDXVH RIWKH VWDWLVWLFDO LQGHSHQGHQFH RI = DQG =WKHUWKQRQFHQWUDOPRPHQWRI<PD\EHGHULYHGIURPWKHPHDQVRI J DQG RI J ZKHUH LQ WKH H[SUHVVLRQ IRU WKH PHDQ RI J ZH UHSODFH ^α µ2, σε2} E\ ^Uα, Uµ2, Uσε2} DQG VLPLODUO\ LQ WKH H[SUHVVLRQ IRU WKH PHDQRIJZHUHSODFHσε2E\Uσε27KHUWKQRQFHQWUDOPRPHQWRI<LV ω @U`(^>J = ω ω @U` ( J = ω (^H[S^ Uαλ > σε1= λ±@Uµ2 Uσε2ρ=`` [(^H[S>Uσε2 ρ =@`
7KH IXQFWLRQ J LV D ORJQRUPDO YDULDEOH DQG ZH KDYH IRU WKH UWK PRPHQWRIJ H[DFW
ω @U` H[S> ρ Uσε2)2/] (^>J = ω
UHIHUWR-RKQVRQ.RW]DQG%DODNULVKQDQ &KDSWHURUWR6WXDUW DQG2UG 6HFWLRQ ([SDQGLQJ J LQWR D 7D\ORU VHULHV LQ WHUPV RI SRZHUV RI = DURXQG ]HURDQGWDNLQJH[SHFWDWLRQRIWKHILUVWIRXUWHUPVZHREWDLQIRUWKHUWK PRPHQWRIJ (^>J = ω @U`≅H[S Uµ2)
[{ > Uα)(λ−1)σε12 + U ασε1+ρσε2)2]`
&+$37(5 7+(500(5525',675,%87,21
)URP DQG DQ H[SOLFLW DSSUR[LPDWH H[SUHVVLRQ IRU WKH UWKQRQFHQWUDOPRPHQWRI<LV ( Uµ ρ Uσε2)2/)] [{ > Uα Ҟ(λ−1)җσε12+ U ασε1+ρσε2)2]}.
)URP WKH QRQFHQWUDO PRPHQWVWKH VNHZQHVV DQG NXUWRVLV PHDVXUHV FDQEHFDOFXODWHGE\ 6WXDUWDQG2UG S ¥β (> <µ @σ >µ µ µ (µ @σ β (> <µ @σ >µ µ µ µ µ (µ @σ± )RUWKHQRUPDOGLVWULEXWLRQERWKPHDVXUHVDUH]HUR (TXDWLRQV DQG FDQQRZEHXVHGWRH[SORUHDUHDVLQWKH ¥ββ SODQHZKHUHWKH500HUURUGLVWULEXWLRQKDVDUHDOL]DWLRQDQG WKHUHIRUH PD\ UHSUHVHQW GLVWULEXWLRQV VKDULQJ WKH VDPH VNHZQHVV DQG NXUWRVLV YDOXHV 6LQFH Uµ LQ LV D VFDOH SDUDPHWHU LW GRHV QRW LQIOXHQFH HLWKHU WKH VNHZQHVV √β1) RU WKH NXUWRVLV β2 PHDVXUHV ERWK DUHVFDOHLQYDULDQW 7KHVKDSHRIWKH500HUURUGLVWULEXWLRQLVWKHUHIRUH GHWHUPLQHGRQO\E\WKHILYHSDUDPHWHUV^αλρσεσε` /HWWKHSDUDPHWHUλLQ EHUHH[SUHVVHGDV λ λσε7KLVDOORZV XVLQJ YDOXHV RI λ0 WKDW DUH QRW LQIOXHQFHG E\ WKH VFDOH RI σε1 IXUWKHU MXVWLILFDWLRQFDQEHIRXQGLQ6KRUH )RU DQ\ DUELWUDULO\ VHOHFWHG YDOXHV RI WKH HUURU VWDQGDUG GHYLDWLRQV ^σεσε`ZHQRZZLVKWRH[SORUHWKHHIIHFWWKDW^αλ0, ρ`KDYHRQWKH VKDSH RI WKH UHVXOWLQJ HUURU GLVWULEXWLRQ $WWHPSWLQJ WR REWDLQ SORWV RI UHVSRQVH VXUIDFHV IRU ^√β1, β` LQ WHUPV RI DQ\ WZR RI WKH DERYH SDUDPHWHUV SURYHG WR EH FRPSXWDWLRQDOO\ SURKLELWLYH 7KHUHIRUH YDOXHV RI ^√β1, β` IRU VHOHFWHG YDOXHV RI ^α λ0` DUH JLYHQ IRU ρ LQ 7DEOH DQG IRU ρ LQ 7DEOH 7KH YDOXHV LQ WKHVH WDEOHV PD\ KHOS DVVHVVKRZWKHYDULRXVSDUDPHWHUVLQIOXHQFHVKDSHFKDUDFWHULVWLFVRIWKH 500HUURUGLVWULEXWLRQ 7KHHUURUVWDQGDUGGHYLDWLRQV σεDQGσε ZHUHDUELWUDULO\VHOHFWHG +RZHYHUZHFRQVLGHUHGLQVHOHFWLQJWKHVHYDOXHVUHVXOWVREWDLQHGZKHQ WKH ³RULJLQ´ ,17 ZDV ILWWHG WR VRPH NQRZQ GLVWULEXWLRQV 6KRUH DQG WDEOHV WKHUHLQ 7KH YDOXHV VHOHFWHG DUH σε σε ,W LV
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH 9DOXHVRI√β1 XSSHUYDOXHV DQGβ2 IRUVHOHFWHG^αλ`ρ= λ0 −1.0 −0.6 −0.2 0.2 0.6 1.0
−0.6 276.9 90337 7.425 170.9 0.6296 4.135 −0.1588 3.187 −0.6731 3.121 −1.440 4.936
α 0.2 −2.135 15.56 −0.8331 10.89 −0.6385 13.30 −0.5896 13.82 −0.4197 12.45 0.3566 11.77
−0.2 56.55 6054 0.3948 13.74 −0.5023 12.93 −0.6206 13.86 −0.6567 12.58 −0.8360 10.31
0.6 −2.422 9.829 −1.372 5.306 −0.4376 3.024 0.1166 3.304 1.083 5.620 4.974 64.43
1.0 −2.031 5.639 −1.362 3.531 −0.3918 0.9473 0.4742 1.257 1.975 9.089 11.17 300.3
7DEOH 9DOXHVRI√β1 XSSHUYDOXHV DQGβ2 IRUVHOHFWHG^αλ`ρ= λ0 −1.0 −0.6 −0.2 0.2 0.6 1.0
−0.6 297.2 104354 15.25 529.6 −10.93 616.1 −29.57 1921 −7.771 233.7 −4.605 39.46
−0.2 65.97 7763 −0.2718 28.93 −0.5842 23.20 −1.024 22.00 −1.612 25.68 −2.192 33.02
α 0.2 −1.326 5.475 −0.5111 2.222 −0.1232 1.863 0.1560 2.002 0.5202 2.531 1.306 6.120
0.6 −1.564 4.208 −0.7926 1.6316 −0.1087 0.4309 0.4710 0.8398 1.322 4.053 4.280 46.76
1.0 −1.389 2.616 −0.7892 1.160 −0.06455 0.1301 0.6733 0.9625 1.944 8.062 10.12 258.1
&+$37(5 7+(500(5525',675,%87,21
FXUUHQWO\ QRW FRPSOHWHO\ FOHDU KRZ WKHVH SDUDPHWHUV DIIHFW WKH VKDSH SDUDPHWHUV VNHZQHVVDQGNXUWRVLV ,WLVKRZHYHUWREHH[SHFWHGWKDWDV WKHHUURUε1= σε=EHFRPHVVPDOOQDPHO\ε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η IRU WKHVH PRGHOV WKH FRUUHVSRQGLQJ YDULDWLRQV RI WKH 500 HUURU TXDQWLOH IXQFWLRQFDQEHGHULYHG 0RGHO7KHEDVLFPRGHO < H[S^ αλ > 1ε λ±@µε`
0RGHO'LIIHUHQWSUHVHQWDWLRQRIε2 < H[S^ α/λ)[(1+ε1)λ − 1] + µ2 }(1+ ε2).
D
0RGHO'LIIHUHQWSUHVHQWDWLRQRIε1 < H[S^ α/λ)[H[S λε1)±@µ2 + ε2`
E
0RGHO'LIIHUHQWSUHVHQWDWLRQRIERWKε1DQGε2 < H[S^ α/λ)[H[S λε1 ±@µ2 ` ε2)
F
5(63216(02'(/,1*0(7+2'2/2*<
)RUHDFKPRGHOLQWURGXFH (LWKHUIRUεε2: ε2 = σε2[ρ= ρ =@
ZKHUHε1= σε1=DQG=DQG=DUHXQFRUUHODWHG RUIRUεε1: ε1 = σε1[ρ= ρ =@
ZKHUHε2= σε2=DQG=DQG=DUHXQFRUUHODWHG
7KHVHYDULDWLRQVH[WHQGWKHVSHFWUXPRIUHDOOLIHVFHQDULRVWKDW500 FDQPRGHO,QSDUWLFXODUYDULRXVFXUUHQWPRGHOVRIUDQGRPYDULDWLRQDUH VKRZQLQ6HFWLRQWREHVSHFLDOFDVHVRIVRPHRIWKHVHYDULDWLRQV 5HIHUHQFHV >@ -RKQVRQ 1 / .RW] 6 %DODNULVKQDQ 1 &RQWLQXRXV 8QLYDULDWH 'LVWULEXWLRQV²9ROXPHQGHGLWLRQ-RKQ:LOH\ 6RQV >@ 6KRUH + *HQHUDO FRQWURO FKDUWV IRU YDULDEOHV ,QWHUQDWLRQDO -RXUQDO RI 3URGXFWLRQ5HVHDUFK >@6KRUH+ 5HVSRQVH0RGHOLQJ0HWKRGRORJ\ 500 ([SORULQJWKHLPSOLHG HUURU GLVWULEXWLRQ &RPPXQLFDWLRQV LQ 6WDWLVWLFV 7KHRU\ DQG 0HWKRGV >@ 6WXDUW $ 2UG - . .HQGDOO¶V $GYDQFHG 7KHRU\ RI 6WDWLVWLFV 9 'LVWULEXWLRQ7KHRU\&KDUOHV*ULIILQ &RPSDQ\/WG/RQGRQ
&KDSWHU
)LWWLQJ3URFHGXUHV IRUWKH(UURU'LVWULEXWLRQ
,QWURGXFWLRQ $ ILWWLQJ SURFHGXUH DVVLJQV YDOXHV WR WKH SDUDPHWHUV RI D ILWWHG GLVWULEXWLRQ VR WKDW LW GHOLYHUV WKH EHVW SRVVLEOH UHSUHVHQWDWLRQ IRU WKH DSSUR[LPDWHGGLVWULEXWLRQ)RUH[DPSOHLQVWDWLVWLFDOSURFHVVFRQWUROWKH S FRQWURO FKDUW PRVW RIWHQ XVHG WR PRQLWRU WKH SHUFHQWDJH QRQ FRQIRUPDQFHRIDSURFHVV LVWKHUHVXOWRIILWWLQJDQRUPDOGLVWULEXWLRQWR WKH ELQRPLDO GLVWULEXWLRQ7KHIRUPHU LVWKHILWWHG GLVWULEXWLRQ ZKLFKLV WKHSODWIRUPRQZKLFKDOORI6KHZKDUWFRQWUROFKDUWVDUHEDVHG7KHODWWHU WKHELQRPLDO LVWKHWUXHGLVWULEXWLRQRIWKHPRQLWRULQJVWDWLVWLFWKDWLV WKHSHUFHQWDJHREVHUYHGLQDVDPSOHRILWHPV 7KH XVH RI WKH QRUPDO GLVWULEXWLRQ LQ DOO RI 6KHZKDUW FRQWURO FKDUWV SURYLGHV D XQLIRUPLW\ RI SUDFWLFH WKDW HQDEOHV SUDFWLWLRQHUV WR XVH WKH VDPH URXWLQH IRU FRQVWUXFWLQJ FRQWURO FKDUWV HYHQ WKRXJK WKH\ XVH PRQLWRULQJ VWDWLVWLFV ZLWK GLIIHUHQWO\ VKDSHG GLVWULEXWLRQV OLNH WKH 3RLVVRQ WKH ELQRPLDO RU VRPH FRQWLQXRXV GLVWULEXWLRQV WKDW DUH QRW H[WUHPHO\DV\PPHWULFDO VNHZHG ,Q RWKHU VFHQDULRV RQH PD\ ZLVK WR XVH GLVWULEXWLRQ ILWWLQJ EHFDXVH WKHWUXHXQGHUO\LQJGLVWULEXWLRQLVXQNQRZQDQGWKHUHLVDQHHGWRXVHRQH WKDW ZRXOG UHSUHVHQW ZHOO WKH XQNQRZQ GLVWULEXWLRQ ,W LV FRPPRQ SUDFWLFH LQ VXFK FDVHV WR DSSUR[LPDWH WKH ODWWHU E\ ILWWLQJ D SDUDPHWHU ULFKGLVWULEXWLRQZKLFKLVIOH[LEOHHQRXJKWRUHSUHVHQWDZLGHVSHFWUXP RI GLYHUVHO\VKDSHG GLVWULEXWLRQV 7KLV LV WKH FDVH IRU H[DPSOH ZLWK &OHPHQWV PHWKRG IRUSURFHVVFDSDELOLW\DQDO\VLVZKLFKDWWHPSWV WRUHSUHVHQWWKHWUXH RIWHQXQNQRZQ SURFHVVGLVWULEXWLRQE\DPHPEHU
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5),77,1*352&('85(6
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
5(63216(02'(/,1*0(7+2'2/2*<
] γηORJ \ε η !∞ γ,ε ∞\!ε
ZKHUH^ γ, η, ε`DUHUHDOYDOXHGSDUDPHWHUVWKDWQHHGWREHGHWHUPLQHG 7KLV TXDQWLOHIXQFWLRQ WUDQVIRUPV WKH TXDQWLOH RI D UDQGRP YDULDEOH < LQWRWKHFRUUHVSRQGLQJTXDQWLOHRIWKHVWDQGDUGQRUPDOYDULDEOH=ZKHUH ) \ Φ ] DQG ) DQG Φ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Φ ] LV GHVLUHG IRU DOO SRVVLEOH SDLUVRITXDQWLOHV^\]`WKDWVDWLVI\WKHILWWHG ,IWKDWHTXDOLW\ZDV H[DFW IRU DOO SDLUV ZRXOG EH DQ H[DFW UHSUHVHQWDWLRQ IRU WKH TXDQWLOHUHODWLRQVKLSEHWZHHQ<DQG=6LQFHWKHWUXHGLVWULEXWLRQRI<LV QRWJRYHUQHGE\ RWKHUZLVHQRILWWLQJZRXOGEHQHHGHG WKHDERYH HTXDOLW\KROGVRQO\DSSUR[LPDWHO\ 7KHPRUHSDUDPHWHUVDILWWHGGLVWULEXWLRQKDVWKHEHWWHULWVIOH[LELOLW\ LQUHSUHVHQWLQJDZLGHVSHFWUXPRIGLIIHUHQWO\VKDSHGGLVWULEXWLRQV7KLV DVVHUWLRQLVWUXHLUUHVSHFWLYHRIZKLFKDSSURDFKZHFKRRVHIRUWKHILWWLQJ SURFHGXUH $OVR FRYHUDJH LQ WKH √β1,β2) SODQH DQ LVVXH WKDW ZH KDYH UHODWHG WR LQ 6HFWLRQ LV FORVHO\ OLQNHG WR WKH FKRLFH RI WKH ILWWHG GLVWULEXWLRQ $ GLVWULEXWLRQ ZLWK JRRG FRYHUDJH SURSHUWLHV ZLOO DOVR EH YHUVDWLOH HQRXJK WR GHOLYHU JRRG UHSUHVHQWDWLRQ WR GLIIHUHQWO\ VKDSHG GLVWULEXWLRQV
&+$37(5),77,1*352&('85(6
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±H[S>λ \D @\!D DQGNQRZQSDUDPHWHUV^Dλ`,WLVHDV\WRVKRZWKDWWKHPHDQLV λD DQG WKH YDULDQFH LV λ2. :H DSSUR[LPDWH WKLV GLVWULEXWLRQ E\ ILWWLQJ D QRUPDOGLVWULEXWLRQZLWKSDUDPHWHUV^µ, σ} 3XUVXLQJWKHPRPHQWPDWFKLQJDSSURDFKWKHILWWLQJSURFHGXUHUHVXOWV LQWKHIROORZLQJSDUDPHWHUVIRUWKHQRUPDOGLVWULEXWLRQ PDWFKLQJRIWKH PHDQDQGWKHYDULDQFH
σ λµ λD
%\WKHSHUFHQWLOHPDWFKLQJDSSURDFKZHZLVKWRLGHQWLI\YDOXHVIRU WKHSDUDPHWHUVRIWKHQRUPDOGLVWULEXWLRQ WKHILWWHGGLVWULEXWLRQ WKDWZLOO DOORZ XV WR REWDLQ JRRG DSSUR[LPDWLRQ WR TXDQWLOHV RI WKH H[SRQHQWLDO GLVWULEXWLRQ 7KH TXDQWLOH UHODWLRQVKLS EHWZHHQ WKH WZR GLVWULEXWLRQV LV JLYHQE\ Φ ]S ) \S ±H[S>λ \SD @ 3
ZKHUH]S [Sµ σDQG[SLVWKH3WKTXDQWLOHRIWKHUY;ZKLFKKDVD QRUPDOGLVWULEXWLRQZLWKSDUDPHWHUV^µ,σ} WKHGLVWULEXWLRQWKDWZHZLVK WRILW 7R LPSOHPHQW TXDQWLOH PDWFKLQJ ZH GHULYH D VDPSOH RI SDLUV RI FRUUHVSRQGLQJ TXDQWLOHV IURP WKH WZR GLVWULEXWLRQV QDPHO\ TXDQWLOHV WKDWKDYHWKHVDPHYDOXHIRUWKHLUUHVSHFWLYH&'),QRXUFDVHZHGHULYH SDLUVRITXDQWLOHV^]S\S`ZKHUH]SDQG\SDUHWKH3WKSHUFHQWLOHVRI WKH VWDQGDUG QRUPDO DQG WKH H[SRQHQWLDO GLVWULEXWLRQV UHVSHFWLYHO\ 2EWDLQLQJ WKHVH TXDQWLOH YDOXHV LV HDV\ )LUVW GHWHUPLQH D YDOXH RI \ 7KHQFDOFXODWHIURPWKHNQRZQH[SRQHQWLDOGLVWULEXWLRQLWV&')YDOXH
5(63216(02'(/,1*0(7+2'2/2*<
3>XVH @)LQDOO\ILQGWKHFRUUHVSRQGLQJVWDQGDUGQRUPDOTXDQWLOH 2XU PRGHO IRU WKH TXDQWLOH RI WKH H[SRQHQWLDO GLVWULEXWLRQ WKH DSSUR[LPDWHG GLVWULEXWLRQ H[SUHVVHG LQ WHUPV RI WKH ILWWHG QRUPDO GLVWULEXWLRQ WKHDSSUR[LPDWLQJGLVWULEXWLRQ LV
\S≅[S µσ]S
/LQHDUUHJUHVVLRQPD\EHXVHGWRILQGYDOXHVRIµDQGσ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
&+$37(5),77,1*352&('85(6
H[SOLFLW IRUP VD\ ZLWK QR QHHG IRU QXPHULFDO LQWHJUDWLRQ WKH ILWWLQJ SURFHGXUH LQYROYHV VROXWLRQ RI P SUREDEO\ QRQOLQHDU HTXDWLRQV YLD VRPH URRWILQGLQJ URXWLQH ,Q RWKHU FDVHV QR H[SOLFLW FORVHGIRUP HTXDWLRQVIRUWKHPRPHQWVLQWHUPVRIWKHGLVWULEXWLRQ VSDUDPHWHUVDUH DYDLODEOH )RU H[DPSOH VRPH RU DOO RI WKH P HTXDWLRQV PD\ UHTXLUH QXPHULFDO LQWHJUDWLRQ ,Q WKHVH FDVHV UHVRUWLQJ WR VRPH PRUH FRPSOLFDWHGQXPHULFDOURXWLQHVZLOOEHQHHGHGSRVVLEO\LQYROYLQJVRPH DSSUR[LPDWLRQV ,Q &KDSWHU H[DFW H[SUHVVLRQV WR FDOFXODWH 500 PRPHQWV ZHUH GHYHORSHG6LQFHQRFORVHGIRUPIRUPXODHZHUHREWDLQDEOHDSSUR[LPDWH H[SUHVVLRQVZHUHGHULYHGEDVHGRQD7D\ORUH[SDQVLRQRIWKHJIXQFWLRQ ,Q &KDSWHU ZH KDYH VLPLODUO\ GHULYHG DQ H[SOLFLW DSSUR[LPDWH H[SUHVVLRQ IRU WKH GI RI WKH ORJWUDQVIRUPHG UHVSRQVH : EDVHG RQ D 7D\ORU VHULHV H[SDQVLRQ RI WKH FRQGLWLRQDO GI RI : %RWK HTXDWLRQV PD\ EH XVHG LQ D PRPHQWPDWFKLQJ SURFHGXUH +RZHYHU HPSOR\LQJ WKH DSSUR[LPDWH GI VWLOO UHTXLUHV QXPHULFDO LQWHJUDWLRQ WR FDOFXODWH WKH PRPHQWV FRXSOHG ZLWK D URRWLGHQWLI\LQJ URXWLQH 7KHVH PHWKRGV DUH RSWLRQDO DQG ZLOO QRW EH SXUVXHG LQ WKLV FKDSWHU RWKHU PHWKRGVWKDWDUHQXPHULFDOO\HDVLHUWRLPSOHPHQWDUHGHYHORSHG $QRWKHU SUREOHP HQFRXQWHUHG LQ DWWHPSWLQJ WR ILW WKH 500 HUURU GLVWULEXWLRQLVWKHUHODWLYHO\ODUJHQXPEHURISDUDPHWHUVLQWKHPRGHO,Q IDFWWRILWWKH500HUURUGLVWULEXWLRQZHQHHGWRGHWHUPLQHWKHYDOXHVRI VL[SDUDPHWHUV^α, λ, µ2, ρ, σε1, σε2}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η ZHREWDLQ
5(63216(02'(/,1*0(7+2'2/2*<
< H[S^ αλ > 1σε1= λ±@µσε2[ρ= ρ =@` λ
: ORJ < αλ > 1σε1= ±@µσε2[ρ= ρ
=@
,W LV HDV\ WR UHDOL]H WKDW LQWURGXFLQJ ρ=± DFKLHYHV WKUHH GHVLUDEOH JRDOV )LUVW ZHHOLPLQDWHRQH SDUDPHWHU ρ IURP WKH VHW RI SDUDPHWHUV WKDW QHHG WR EH GHWHUPLQHG 6HFRQGO\ = GLVDSSHDUV IURP WKH TXDQWLOH IXQFWLRQ 7KLV WXUQV WKH 500 TXDQWLOH IXQFWLRQ LQWR DQ ,QYHUVH 1RUPDOL]LQJ 7UDQVIRUPDWLRQ ,17 QDPHO\ DQ H[SUHVVLRQ WKDW GHOLYHUV WKHTXDQWLOHRIWKH500HUURUGLVWULEXWLRQLQWHUPVRIWKHTXDQWLOHRID VLQJOH UY WKH VWDQGDUG QRUPDO = $V D UHVXOW WKH SDUDPHWHU µ2 QRZ UHSUHVHQWV WKH ORJ RI WKH UHVSRQVH PHGLDQ 0 WKDW LV µ2= ORJ 0 IRU = < 0 7KH SDUDPHWHU µ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µ σ ={ αλ > σε1= λ±@µσε2ρ=µ} / σ >σε2 ρ /σ ]=
ω1) + 462(Ζ2; ω 46 = ω
ZKHUH µ DQG σ DUH WKH JLYHQ H[DFW PHDQ DQG VWDQGDUG GHYLDWLRQ UHVSHFWLYHO\RI:46DQG46DUHVHSDUDWHH[SUHVVLRQVLQWHUPVRI=
CHAPTER 10 FITTING PROCEDURES
169
and Z2, respectively, and ω1= {α, λ, µ2, σε1, σε2ρ}, ω2= {σε2(1-ρ2)(1/2)}, are the parameters that need to be determined (six parameters altogether). Note that QS1 and QS2 differ from Q1 and Q2, respectively, given in (9.16). Since Z1 and Z2 are independent standard normal variables, the r-th moment of WS is Mr(WS) = E[(WS)r] =
r
r
j =0
∑ j M
j
(QS1 ) M r − j (QS 2 ) , (10.10)
where Mj(QS1) and Mr-j(QS2) are the non-central j-th and (r-j)-th moments of QS1 and of QS2, respectively. For the latter we have Mk(QS1) = E[(QS1)k] =
∫
∞
−∞
{{(α/λ)[(η+σε1z)λ – 1] + µ2 + σε2ρz - µ}/σ}k φ(z)dz , (10.11) Mk(QS2) =
∫
∞
−∞
[(σε2/σ)(1-ρ2)(1/2) z]k φ(z)dz
= [(σε2/σ)(1-ρ2)(1/2)]k µk(Z),
(10.12)
where µk(Z) is the k-th moment of the standard normal distribution. For k odd, we have identically µk(Z)=0. For k even, we have for the first few moments µ2(Z) = E(Z2) =1, µ4(Z) = 3, µ6(Z) = 15, µ8(Z) = 105. This implies that to calculate a moment of any desirable degree via (10.10) only numerical integration of (10.11) is needed. To identify the parameter set, {ω1, ω2}, the following routine needs to be applied: Find {ω1, ω2} that minimize the objective function 6
OF = [M1 (WS)]2 +
∑
[Mr (WS)/µr(WS) - 1]2,
(10.13)
2
where µr(WS)= E[(WS)r] is the given exact r-th moment of WS (the standardized W), namely, the r-th moment associated with the distribution that we wish to represent by the RMM model (these moments may be derived from available explicit expressions for the moments, or numerically calculated with the exact density function).
5(63216(02'(/,1*0(7+2'2/2*<
1RWH WKDW VLQFH :6 LV WKH VWDQGDUGL]HG YDULDEOH 0 :6 DQG 0 :6 DUHH[SHFWHGWREH]HURDQGUHVSHFWLYHO\$OVRQRWHWKDWVLQFH WKHUH DUH VL[ SDUDPHWHUVWKH VXP H[WHQGV RYHUWKH ILUVW VL[ PRPHQWV LQ RUGHU WR REWDLQ D XQLTXH VROXWLRQ 0DWFKLQJ D VPDOOHU QXPEHU RI PRPHQWVWKHREWDLQHGVROXWLRQPD\QRWEHXQLTXH $OWHUQDWLYHO\WRPDWFKRQO\WKHILUVWIRXUPRPHQWVLQWURGXFHρ ± µ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ρ WKHQ DTXDQWLOHPD\EHHDVLO\FDOFXODWHGIURP E\IROORZLQJWKHSDWK 'HWHUPLQH3!)LQG]3!)LQG\3IURP ZKHUH]3LVWKH3WKSHUFHQWLOHRIWKHVWDQGDUGQRUPDOGLVWULEXWLRQ 7UDGLWLRQDO FULWHULD IRU JRRGQHVVRIILW OLNH FRUUHODWLRQ VWDQGDUG GHYLDWLRQ RI WKH HUURUV RU 0LQLPXP GLVWDQFH PHDVXUHV WR DVVHVV GLVWULEXWLRQDOILW UHIHUIRUH[DPSOHWR:DULFN ,Q DGGLWLRQ WR WKHVH JRRGQHVVRIILW FULWHULD YDULRXV HUURU SORWV PD\ EHHPSOR\HGWRGHPRQVWUDWHWKHUHVXOWDQWJRRGQHVVRIILW)RUH[DPSOHD SORWRIWKHHUURURIWKHUHVSHFWLYHGHQVLW\IXQFWLRQV ILWWHGYVH[DFW RUD SORW RI WKH HUURU RI WKH TXDQWLOH YDOXHV $OO FRPSDULVRQV PD\ EH PDGH
&+$37(5),77,1*352&('85(6
HLWKHUZLWKUHJDUGWRWKHRULJLQDOVFDOH < RUWRWKHORJWUDQVIRUPHGVFDOH : )RU D JRRG VRXUFH UHODWLQJ WR FRPSDULVRQV RI ILWWHG PRGHOV UHIHU WR %XUQKDPDQG$QGHUVRQ )LWWLQJYLD4XDQWLOH0DWFKLQJ &RQVLGHU WKH UHVSRQVH < ZLWK D VSHFLILHG &') * \ )URP WKLV GLVWULEXWLRQQTXDQWLOHVDUHGUDZQZLWKWKHLUFRUUHVSRQGLQJ&')YDOXHV QDPHO\WKHVDPSOHFRQVLVWVRI^\L* \L ` L Q 9DOXHV IRU WKH SDUDPHWHUV RI WKH 500 TXDQWLOH IXQFWLRQ DUH GHVLUHGVRWKDWWKHREMHFWLYHIXQFWLRQ
2)
Q
L
\L±\IL
EHPLQLPL]HGZKHUH^\IL`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
: αλ > 1σε1= λ±@µσε2ρ=
7KLV LV D TXDQWLOH IXQFWLRQ ZLWK D VLQJOH VWDQGDUG QRUPDO UY = ZKLFKFDQQRZEHILWWHGYLDD4XDQWLOH0DWFKLQJSURFHGXUH7KHVDPH UHVXOWZRXOGKDYHEHHQREWDLQHGE\VHWWLQJLQ ρ 1RWHWKDW
5(63216(02'(/,1*0(7+2'2/2*<
LPSOLHV WKDW WKH FRHIILFLHQWV RI = DV REWDLQHG IURP WKH ILWWLQJ SURFHGXUHPD\KDYHGLIIHUHQWVLJQV WKDWLVLIρ $QHTXLYDOHQWIRUPIRU LV>UHIHUWR @ : ORJ < αλ ^>1σε1[ρ= ρ =@@λ±`µσε2= ,QWURGXFLQJρ ±ZHREWDLQFRUUHVSRQGLQJWR
: ORJ < αλ > 1σε1ρ= λ±@µσε2=
/LNH ZLWK WKLV DJDLQ LPSOLHV WKDW DV D UHVXOW RI GLVWULEXWLRQ ILWWLQJ WKH FRHIILFLHQWV RI = PD\ KDYH GLIIHUHQW VLJQV WKDW LV LI ρ 6LQFHLQSUDFWLFHRQHFDQQRWWHOOZKHWKHU RU LVILWWHG WKH\ DUHHTXLYDOHQW LWFDQEHH[SHFWHGWKDWHLWKHURIWKHFRHIILFLHQWVRI= EXW QRWERWK PD\WXUQRXWWREHQHJDWLYH)XUWKHUPRUHLIUHTXLUHGWRSURYLGH LQLWLDOSDUDPHWHUYDOXHVIRUWKHILWWLQJSURFHGXUHQHJDWLYHYDOXHVIRUWKH FRHIILFLHQWVRI=VKRXOGQRWEHUXOHGRXW :H DUH QRZ UHDG\ WR LPSOHPHQW D 4XDQWLOH 0DWFKLQJ SURFHGXUH 7KHTXDQWLOHIXQFWLRQ> RU @KDVRQO\ILYHSDUDPHWHUVWREH GHWHUPLQHG7RSUHVHUYHWKHPHGLDQRIWKHDSSUR[LPDWHGGLVWULEXWLRQZH PD\ZLVKWRVHWµ2= ORJ 0 ZKHUH0LVWKH H[DFW PHGLDQ7KLVOHDYHV XV ZLWK IRXU SDUDPHWHUV WKDW QHHG WR EH GHWHUPLQHG ^α, λ, σε1 ±σε2` 7KH QXPEHU RI GHJUHHV RI IUHHGRP LV ODUJH HQRXJK WR SUHVHUYH WKH IOH[LELOLW\QHHGHGLQUHSUHVHQWLQJDZLGHVSHFWUXPRIGLIIHUHQWO\VKDSHG GLVWULEXWLRQV $1//6SURFHGXUHPD\EHDSSOLHGWRILW RU WRWKH JLYHQVHWRIQSDLUVRITXDQWLOHYDOXHV^]LORJ \L `ZKHUH
3L ):>ORJ \L ] = Φ ]L , L Q
DQG):LVWKHWUXH H[DFW &')RI: ORJ < QDPHO\
):>ORJ \ ] = * \
'HULYDWLRQRIWKHVHWRITXDQWLOHVWREHXVHGLQWKHILWWLQJSURFHGXUH LVGHVFULEHGE\WKHIROORZLQJSDWK 6HOHFW\3Æ)LQG&')YDOXH3 * \3 Æ,GHQWLI\]3 ZKHUH]3LVWKH3WKTXDQWLOHRIWKHVWDQGDUGQRUPDOGLVWULEXWLRQ 6HFWLRQGHPRQVWUDWHVLPSOHPHQWDWLRQRIWKHWZRILWWLQJPHWKRGV WRDJDPPDGLVWULEXWLRQ,Q&KDSWHU WKHWZRH[DPSOHVJLYHQKHUHDUH H[SDQGHGWRDZLGHUVDPSOHRIGLVWULEXWLRQV
&+$37(5),77,1*352&('85(6
7ZR1XPHULFDO([DPSOHV $PRPHQWPDWFKLQJH[DPSOH 7KH PRPHQWPDWFKLQJ ILWWLQJ SURFHGXUH H[SRXQGHG LQ 6HFWLRQ LV XVHG WR ILW WKH 500 HUURU GLVWULEXWLRQ WR WKH JDPPD GLVWULEXWLRQ 7KHODWWHUKDVIRUWKHILUVWIRXUPRPHQWV PHDQYDULDQFHVNHZQHVVDQG NXUWRVLV DQGWKHORJPHGLDQ /0HG µ σ2 = 6N .X /0HG 1RWHWKDWWKHVNHZQHVVDQGNXUWRVLVPHDVXUHVDUHGHILQHGE\ 6N µ σ.X µσ4
)RU WKH ORJWUDQVIRUPHG JDPPD ORJ < ZH REWDLQ E\ QXPHULFDO LQWHJUDWLRQWKHH[DFWILUVWVL[VWDQGDUGL]HGPRPHQWVJLYHQLQ7DEOH ,PSOHPHQWLQJ WKH PLQLPL]DWLRQ URXWLQH WKH IROORZLQJ SDUDPHWHUV YDOXHVDUHGHULYHG α λ0 = λσε1 = µ2 =
σε1 = σε2 = ρ =
7KH λ LQ ZDV UHSODFHG E\ λ0/σε1 LQ RUGHU WR HQVXUH WKDW VPDOO YDOXHVZLOOEHREWDLQHG ,W LV RI LQWHUHVW WR QRWH ρ. ,WV YDOXH ZDV IRXQG LQ WKH ERXQGHG UDQJH ZKHUHLWVYDOXHLVH[SHFWHGWREHHYHQWKRXJKQRUHVWULFWLRQZDV LPSRVHGRQWKHVHDUFKURXWLQH 7DEOHFRPSDUHVWKHILUVWVL[VWDQGDUGL]HGPRPHQWVRIWKHORJRI WKHJDPPDYDULDEOH H[DFW ZLWKWKRVHFDOFXODWHGIURPWKHILWWHGTXDQWLOH IXQFWLRQ 'HYLDWLRQV GR QRW H[FHHG KDOI D SHUFHQW HYHQ WKRXJK YHU\KLJKGHJUHHPRPHQWVDUHFRPSDUHG 7DEOH)LUVWVL[VWDQGDUGL]HGPRPHQWVRIORJJDPPDZLWKSDUDPHWHUVα=3, β=2. 'HJUHHRI0RPHQW N N N N N N
µN :6 H[DFW
0N :6 ILWWHG
'HYLDWLRQ
5(63216(02'(/,1*0(7+2'2/2*<
$TXDQWLOHPDWFKLQJH[DPSOH 7KHTXDQWLOHEDVHGILWWLQJSURFHGXUHH[SRXQGHGLQ6HFWLRQLVXVHG WRILWWKHJDPPD GLVWULEXWLRQ7KHPRPHQWVRIWKLVGLVWULEXWLRQZHUH JLYHQLQWKHSUHYLRXVH[DPSOH 7R FDUU\ RXW WKH ILWWLQJ SURFHGXUH D VDPSOH RI TXDQWLOH YDOXHV LV VHOHFWHGIURPWKHVWDQGDUGQRUPDOGLVWULEXWLRQ ^]L` ^ L`L ZLWK FRUUHVSRQGLQJ TXDQWLOH YDOXHV IURP WKH JDPPD GLVWULEXWLRQ >7KLV URXWLQH JRHV LQ WKH RSSRVLWH GLUHFWLRQ WR WKDW JLYHQ LQ %RWK GLUHFWLRQVDUHHTXDOO\YDOLG@ ,PSRVLQJ µ2 ORJ 0 /0HG , DQG DSSO\LQJ D 1//6 SURFHGXUH >XVLQJ HLWKHU RI WKH HTXLYDOHQW IRUPV RU @ WKH SDUDPHWHUVREWDLQHGDUH α λ0 σε1 σε2)ρ = 7KH UHVLGXDO YDULDQFH IURP WKLV ILWWLQJ LV σUHV (TXDWLQJWKHUHVLGXDOYDULDQFHWRWKHYDULDQFHRIε>UHIHUWR @
σε2)2 ρ
ZHREWDLQIURPWKHODVWH[SUHVVLRQLQ DQGIURP
ρ σε
1RWHWKDWLQWKHILWWHGTXDQWLOHIXQFWLRQZHKDYHDVVXPHGρ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
CHAPTER 10 FITTING PROCEDURES
175
0.0008
0.12
0.0006
0.1
0.0004
0.08
0.0002
0.06 -0.0002
0.04
5
10
15
20
25
-0.0004
0.02
-0.0006 5
10
15
20
25
Figure 10.1. Plots of the density function, exact and fitted (left) and of the error for Gamma (3,2) (based on quantile-matching).
Figure 10.2. Plots of the quantile function (approximate and exact, left) and of the deviation (approximate minus exact, right) for the Gamma (3,2). Horizontal axis is the standard normal quantile (z). Vertical axis (left plot) is the standardized gamma quantile.
5(63216(02'(/,1*0(7+2'2/2*<
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
&KDSWHU
(VWLPDWLQJWKH(UURU'LVWULEXWLRQ
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
< 0 H[S^ αλ > ε1 λ±@ε` λ
D
< 0 H[S^ αλ > ε1 ±@` ε2)
E
< 0 H[S^ αλ >H[S λε1 ±@ε2`
F
< 0 H[S^ αλ >H[S λε1 ±@` ε2
G
ZKHUH D FRUUHVSRQGV WR WKH RULJLQDO 500 PRGHO GHULYHG D[LRPDWLFDOO\LQ&KDSWHU
5(63216(02'(/,1*0(7+2'2/2*<
$VVXPH WKDW WKH HUURUV ε1 σε1= DQG ε σε2= RULJLQDWH LQ D EL YDULDWH QRUPDO GLVWULEXWLRQ ZLWK FRUUHODWLRQ ρ, RU HTXLYDOHQWO\ WKDW = DQG=DUHIURPDELYDULDWHVWDQGDUGQRUPDOGLVWULEXWLRQZLWKFRUUHODWLRQ ρ%\FRQGLWLRQLQJ=_= ]WKHHUURUVFDQEHUHZULWWHQDV ε1 = σε1=
ε2 = σε2[ρ= ρ =@
D
ZKHUH=DQG=DUHQRZLQGHSHQGHQWVWDQGDUGQRUPDOYDULDEOHV 6LPLODUO\E\FRQGLWLRQLQJ=_= ]ZHKDYH ε2 = σε2=
ε1 = σε1[ρ= ρ =@
E
ZKHUH = DQG = DUH LQGHSHQGHQW VWDQGDUG QRUPDO YDULDEOHV (TXDWLRQV DQG JHQHUDWH DOWRJHWKHU SRVVLEOH YDULDWLRQV IRU WKH TXDQWLOHIXQFWLRQRIWKH500HUURUGLVWULEXWLRQ UHIHUWR6HFWLRQ (VWLPDWLRQSURFHGXUHVIRUWZRYHUVLRQVRIWKH500TXDQWLOHIXQFWLRQ DUH GHYHORSHG 7KH RULJLQDO PRGHO D DQG WKH YDULDWLRQ JLYHQ E\ F 7KHUH DUH WKUHH UHDVRQV IRU VHOHFWLQJ F LQ DGGLWLRQ WR D )LUVW WKLV PRGHO KDV RQH OHVV SDUDPHWHU WKDW QHHGV HVWLPDWLRQ 6HFRQGO\LQDQXPHULFDOVHDUFKIRUHVWLPDWHVRIWKHSDUDPHWHUVWKHWHUP σε1=1)λ ZKLFK DSSHDUV LQ D PD\ RFFDVLRQDOO\ DVVXPH LPDJLQDU\ YDOXHV WKH H[SUHVVLRQ LQ EUDFNHWV EHFRPHV QHJDWLYH 7KLV SUREOHP GRHV QRW H[LVW ZKHQ D QXPHULFDO VHDUFK LV FRQGXFWHG IRU WKH SDUDPHWHUVWKDWDSSHDULQ F 7KLUGO\LIWKHQRUPDOHUURUVLQ F DUHDVVXPHGWREHLQGHSHQGHQW ρ ORJ < LVWKHVXPRILQGHSHQGHQW H[SRQHQWLDO DQG QRUPDO YDULDEOHV 7KLV LV D FRQYHQLHQW IRUP IRU WKH 500TXDQWLOHIXQFWLRQZKLFKKDVFHUWDLQGLVWULEXWLRQDODGYDQWDJHV IRU H[DPSOHPRPHQWVDUHHDV\WRFDOFXODWH ,QWURGXFLQJ WKH HUURUV LQ WHUPV RI WKH LQGHSHQGHQW = DQG = IURP LQWR D ZHREWDLQWKHWZRHTXLYDOHQWIRUPV : ORJ < ORJ 0 αλ > σε1= λ±@σε2[ρ= ρ =@
D
: ORJ 0 αλ ^^σε1[ρ= ρ =@`λ±`σε2= E 6LPLODUO\IRU F WKHWZRHTXLYDOHQWIRUPVDUH
&+$37(5(67,0$7,1*7+((5525',675,%87,21
: ORJ 0 αλ >H[S λσε1= ±@σε2[ρ= ρ =@ F : ORJ 0 αλ ^H[S^λσε1[ρ= ρ =@`±`σε2= G 1RWH WKDW HTXLYDOHQW IRUPV LPSO\ WKDW LQ HVWLPDWLQJ D EDVHG RQ VDPSOH GDWD ZH FDQQRW WHOO ZKHWKHU ZH ILW D RU E 7R UHDOL]H WKDW LQWURGXFH IRU ρ WKH WKUHH YDOXHV ^ ` 7KH VDPH H[SUHVVLRQV DUH REWDLQHG WKRXJK SRVVLEO\ KDYLQJ GLIIHUHQW SDUDPHWHUV YDOXHV6LPLODUO\LIRQHFKRVHWRILW F LWFDQQRWEHGHWHUPLQHGLQ DGYDQFHZKHWKHU F RU G ZHUHILWWHG *LYHQ WKH ODUJH QXPEHU RI SDUDPHWHUV LQFOXGHG LQ WKH TXDQWLOH IXQFWLRQVRI DWPRVWVL[SDUDPHWHUV RQHGHJUHHRIIUHHGRPFDQ EH JLYHQ XS ZLWKRXW ORVLQJ PXFK LQ WKH IOH[LELOLW\ RI WR GHSLFW GLIIHUHQWO\VKDSHG GLVWULEXWLRQV 6HWWLQJ ρ ± DG EHFRPH UHVSHFWLYHO\
: ORJ 0 αλ > σε1= λ±@σε2ρ=
D
: ORJ 0 αλ > σε1ρ= λ±@σε2=
E
: ORJ 0 αλ >H[S λσε1= ±@σε2ρ=
F
: ORJ 0 αλ >H[S λσε1ρ= ±@σε2=
G
ZKHUH=LVDVWDQGDUGQRUPDOUDQGRPYDULDEOH 5HZULWWHQ ZLWK GLIIHUHQW SDUDPHWHUV D DQG E PD\ EH FRPELQHGWRJLYH
: ORJ < ORJ 0 $ > %= &±@'=
ZKHUH ^0 $ % & '` DUH SDUDPHWHUV WKDW QHHG WREH HVWLPDWHG 1RWH WKDW HLWKHU % RU ' FDQ DVVXPH QHJDWLYH YDOXHV >EXW QRW ERWK UHIHU WR DE @ 6LPLODUO\IURP F DQG G
: ORJ < ORJ 0 Α >H[S %= ±@'=
DQG LQ ILWWLQJ WR VDPSOH GDWD SURYLGHG λ! HLWKHU % RU ' FDQ DVVXPHQHJDWLYHYDOXHV>EXWQRWERWKUHIHUWR FG @ )RU = ERWK DQG JLYH : ORJ 0 ZKHUH 0 LV WKH UHVSRQVHPHGLDQ WKHPHGLDQRI< (VWLPDWLQJSURFHGXUHVDSSOLFDEOHIRUWKHSDUDPHWHUVLQ DQGLQ ZLOO EH GHYHORSHG :H DGGUHVV WZR W\SHV RI HVWLPDWLRQ
5(63216(02'(/,1*0(7+2'2/2*<
DSSURDFKHV (VWLPDWLRQEDVHGRQSHUFHQWLOHPDWFKLQJ 6HFWLRQ DQG HVWLPDWLRQ EDVHG RQ PRPHQWPDWFKLQJ 6HFWLRQ 0D[LPXP OLNHOLKRRGHVWLPDWLRQKDVEHHQGHYHORSHGIRUWKHJHQHUDO500PRGHOLQ &KDSWHUDQGLWPD\EHHDVLO\DGDSWHGIRUHVWLPDWLRQRIWKH500HUURU GLVWULEXWLRQ ,Q WKH IROORZLQJ VHFWLRQV LW LV DVVXPHG WKDW D VDPSOH RI Q REVHUYDWLRQV ^\L` LV DYDLODEOH DQG WKDW WKH REVHUYDWLRQV DUH H[SUHVVHG DOVRLQWHUPVRIWKHORJWUDQVIRUPHGYDOXHV^ZL` 3HUFHQWLOH%DVHG(VWLPDWLRQ 7KHHVWLPDWLRQSURFHGXUH $ SHUFHQWLOHEDVHG PDWFKLQJ PHWKRG LV XVHGWR HVWLPDWH WKH SDUDPHWHUV RI DQG 7KLV LQYROYHV HVWLPDWLQJ UHVSRQVH TXDQWLOHV IURP WKHJLYHQVDPSOH LGHQWLI\LQJFRUUHVSRQGLQJVWDQGDUGQRUPDOTXDQWLOHV DQGWKHQXVLQJQRQOLQHDUOHDVWVTXDUHV 1//6 IRUILWWLQJ 7ZRDSSURDFKHVFDQEHSXUVXHG $ 6HOHFWDVHWRI&')YDOXHV7KHQLGHQWLI\ FDOFXODWH WKHDVVRFLDWHG VWDQGDUG QRUPDO TXDQWLOHV ^]L` DQG HVWLPDWH IURP WKH VDPSOH WKH FRUUHVSRQGLQJUHVSRQVHTXDQWLOHV^\L` % )URPWKHVDPSOH VRUGHUHGREVHUYDWLRQVHVWLPDWHWKHFRUUHVSRQGLQJ &')YDOXHV&DOFXODWHWKHUHVSHFWLYHVWDQGDUGQRUPDOTXDQWLOHV 7KH ILUVW DSSURDFK UHTXLUHV WR GHWHUPLQH DUELWUDULO\ D VHW RI &') YDOXHV ^SL` ZKHUH S ) \ DQG ) \ LV WKH &') RI WKH UHVSRQVH < )URP WKHVH YDOXHV WKH FRUUHVSRQGLQJ VWDQGDUG QRUPDO TXDQWLOHV DUH LGHQWLILHG )LQDOO\ VDPSOH HVWLPDWHV IRU ^\S` DUH GHULYHG QDPHO\ HVWLPDWHVIRUTXDQWLOHVRI<VXFKWKDW3U <\S ) \S S 7KHVHFRQGDSSURDFKUHTXLUHVWRHVWLPDWHIURPWKHVDPSOHWKHYDOXH RI SL ) \ L ZKHUH \ L LV WKH LWK RUGHU VWDWLVWLF WKH VDPSOH YDOXH IRU ZKLFK WKHUH DUH L VPDOOHU YDOXHV :H ZLOO UHIHU WR WKLV GHILQLWLRQ DQG QRWDWLRQ VKRUWO\ )URP WKLV HVWLPDWH WKH UHVSHFWLYH VWDQGDUG QRUPDO TXDQWLOH]LLVFDOFXODWHG
&+$37(5(67,0$7,1*7+((5525',675,%87,21
)RUERWKDSSURDFKHVWKHILQDOVDPSOHIRULPSOHPHQWDWLRQRI1//6LV FRPSULVHGRIWKHRUGHUHGVDPSOHVHW ^ ]\ ]\ ]Q\Q `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`/HWWKHRUGHUVWDWLVWLFVIURPWKLVVDPSOH WKHREVHUYDWLRQV DUUDQJHGLQDQDVFHQGLQJRUGHU EH^\ \ \ Q `>ZHXVHQRWDWLRQIRU RUGHU VWDWLVWLFV WKDW DSSHDUV LQ 2UG DQG 6WDXUW WKRXJK RWKHU QRWDWLRQVH[LVW@1RWHWKDW\ LVWKHVPDOOHVWREVHUYDWLRQDQG\ Q LVWKH ODUJHVW 6XSSRVHWKDWDQHVWLPDWHRIWKHSWKTXDQWLOHLVGHVLUHG QDPHO\WKH YDOXHRIWKHUHVSRQVHIRUZKLFKWKHSUREDELOLW\RIQRWH[FHHGLQJLWLVS (VWLPDWLQJFRPSULVHVWZRVWDJHV ([SUHVV Q SE\UJZKHUHULVDQLQWHJHUDQGJ
5(63216(02'(/,1*0(7+2'2/2*<
)RUH[DPSOHOHWQ S :HKDYH Q S U J (VWLPDWHWKHSWKTXDQWLOH GHQRWHLWE\4S 4S J \ U J\ U U Q $Q H[DPSOH &RQVLGHU WKH VDPSOH RI RUGHUHG REVHUYDWLRQV Q ^`)RUS ZHKDYH Q S 7KHUHIRUH 4 >\ @>\ @ 1RWH WKDW IRU U Q \ Q LV UHSODFHG E\ \ Q ZKLFK LPSOLHV 4S \ Q $OVR QRWH WKDW WKLV PHWKRG LV EDVHG RQ D VLPSOH OLQHDU LQWHUSRODWLRQ EHWZHHQWKHWZRRUGHUVWDWLVWLFVDGMDFHQWWRWKHUHTXLUHGSHUFHQWLOH 7KHUHVSHFWLYHVWDQGDUGQRUPDOTXDQWLOHLVΦ >Φ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
SM M Q
7KH GLIIHUHQFH EHWZHHQ WKH WZR DSSUR[LPDWLRQV ZLOO PRVW RIWHQ EH QHJOLJLEOH)LQGGHWDLOVIRUERWKH[SUHVVLRQVLQ'RGVRQ $QH[DPSOH)RUWKHVDPSOHRIREVHUYDWLRQVJLYHQHDUOLHU ZLWKQ WKH ILUVW RUGHU VWDWLVWLF VPDOOHVW REVHUYDWLRQ LV \ 7KH FRUUHVSRQGLQJ
&+$37(5(67,0$7,1*7+((5525',675,%87,21
&')LVQRWWKHELDVHGYDOXHRIS EXWUDWKHUWKHXQELDVHG YDOXH RI >UHIHU WR @ S ,I WKH DSSUR[LPDWH 0HGLDQ 5DQN ZDV XVHG WKH &') YDOXH REWDLQHG IURP LV S 7KH 0HGLDQ 5DQN WDNHV DFFRXQW RI WKH VDPSOLQJ GLVWULEXWLRQ RI WKH RUGHU VWDWLVWLF DQG LQGHHG HVWLPDWHV DSSUR[LPDWHO\ WKH PHGLDQ RI WKLV GLVWULEXWLRQ WKH H[DFW PHGLDQUDQNFDQEHVKRZQWREHS 7KHUHVSHFWLYHVWDQGDUGQRUPDOTXDQWLOHLVΦ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Φ SM *HQHUDWHWKHVDPSOHIRUWKH1//6ILWWLQJ ^]S\S`ZKHUH]SDQG \SDUHVWDQGDUGQRUPDODQGUHVSRQVHTXDQWLOHVUHVSHFWLYHO\ $ GHWDLOHG QXPHULFDO H[DPSOH WR SUHSDUH D VDPSOH IRU 1//6 &RQVLGHU DJDLQ WKH VDPSOH RI RUGHUHG REVHUYDWLRQV Q ^`8VLQJ ZHKDYHWKHSYDOXHV S ^ ` 7KHFRUUHVSRQGLQJ]YDOXHVDUH ] ^ ` 7KHVDPSOHIRUWKH1//6ILWWLQJLV ^ ]S\S ` ^
`
5(63216(02'(/,1*0(7+2'2/2*<
2QFHDOOYDOXHVRI^]S\S`DUHDYDLODEOH1//6PD\EHLPSOHPHQWHG XVLQJ IRUILWWLQJ HLWKHU RU 8VLQJ WKH VDPSOH PHGLDQ DVDQ HVWLPDWH IRU WKH SDUDPHWHU 0 >S LQ @ PRGHOLQJ ZLWK UHTXLUHVHVWLPDWLQJIRXUSDUDPHWHUVDQGWKUHHLI LVXVHG 2WKHU500YDULDWLRQVPD\VLPLODUO\EHHVWLPDWHG &RPPHQW$UHOHYDQWLVVXHLVWKHUHVSRQVHWRHPSOR\LQWKHHVWLPDWLRQ SURFHGXUH (LWKHU < RU : ORJ < PD\ EH XVHG ,W VHHPV WKDW FRPSXWDWLRQDOFRQYHQLHQFHVKRXOGEHWKHRYHUULGLQJFRQFHUQDQGZLWK WKLVLQPLQGLWLVUHFRPPHQGHGWKDW:EHXVHGDVWKHUHVSRQVH,QWKDW FDVHWKH LQSXWVDPSOH LV ^]S ZS` ZKHUH ZS LV WKH HVWLPDWH RIWKH SWK TXDQWLOHRI: 7KHSHUFHQWLOHEDVHGHVWLPDWLQJSURFHGXUHLVQRZGHPRQVWUDWHGZLWK WZRH[DPSOHVIURP.DULDQDQG'XGHZLF] KHQFHIRUWK.' ZKHUH WKHJHQHUDOL]HG/DPEGDGLVWULEXWLRQZDVXVHGIRUILWWLQJ7KHGDWDIRUWKH H[DPSOHVDUHUHSURGXFHGKHUHZLWKSHUPLVVLRQ 7ZRQXPHULFDOH[DPSOHV SHUFHQWLOHEDVHGHVWLPDWLRQ ([DPSOH%LUWKZHLJKWVRIWZLQV7KH,QGLDQD7ZLQ6WXG\ 7KLVH[DPSOHXVHVGDWDIURPWKH,QGLDQD7ZLQ6WXG\ ILQGGHWDLOVLQ.' S 7KHGDWDFRQVLVWRIELUWKZHLJKWVRIWZLQV^<<`DSSHDULQJLQ 7DEOHLQDQDVFHQGLQJRUGHURI< UHDGOLQHV .'ILWWHGWKHJHQHUDOL]HG/DPEGDGLVWULEXWLRQ>*/'UHIHUWR&KDSWHU DQG WR EHORZ@ XVLQJ PRPHQW PDWFKLQJ :H ZLOO DQDO\]H WKH GLVWULEXWLRQRI7ZLQ < XVLQJ DVWKH500TXDQWLOHIXQFWLRQ )RUWKH<GDWDWKHILUVWIRXUPRPHQWVDUH
µ ( < σ2 9DU <
6N √β (> <µ @σ3
.X (> <µ @σ4
&+$37(5(67,0$7,1*7+((5525',675,%87,21
7DEOH([DPSOH%LUWK:HLJKWVRI7ZLQV ^<
<`
^<
<`
^<
<`
^<
<`
5(63216(02'(/,1*0(7+2'2/2*<
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µ σ2 6N .X 7DEOH([DPSOH5HVXOWVIRUWKHILUVWIRXUDQGWKHODVWIRXURUGHUVWDWLVWLFVRIWKH ELUWKZHLJKWV < RI7ZLQV Q PHGLDQLV 6DPSOH 9DOXH \L SL ]L
L
L
L
L
L
L
L
L
7DEOH([DPSOH(VWLPDWHVDQG &RQILGHQFH,QWHUYDOVIRUWKH500 3DUDPHWHUV GHULYHGIURP1//6 3DUDPHWHU $ % '
(VWLPDWH
$V\PSWRWLF6(
&RQILG,QWHUYDO ^` ^` ^ `
&+$37(5(67,0$7,1*7+((5525',675,%87,21
7KH DYHUDJH RI WKH DEVROXWH YDOXHV RI WKH HUURUV LQ )LJXUH LV *RRGILWLVDFKLHYHG 7KH*/'LQYHUVHGLVWULEXWLRQIXQFWLRQLV < λ λ >3λ 3 λ@3
)LJXUHGLVSOD\VWKHUHODWLYHHUURUVREWDLQHGIURPILWWLQJ*/'E\ PRPHQWPDWFKLQJZLWKSDUDPHWHUV HVWLPDWHV .'S
λ1 λ2 λ λ4
7KH DYHUDJH RI WKH DEVROXWH YDOXHV RI WKH HUURUV LV YV IRU WKH 500 PRGHO 7KH WZR PRGHOV VHHP WR GHOLYHU FRPSDUDEOHJRRGQHVVRIILW.
)LJXUH([DPSOH5HODWLYHHUURUV LQ RIWKHILWWHG YVDFWXDOELUWKZHLJKWV
)LJXUH([DPSOH5HODWLYHHUURUV LQ RIWKHILWWHG*/'YVDFWXDOELUWKZHLJKWV
5(63216(02'(/,1*0(7+2'2/2*<
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` ZKHUH0LVWKHPHGLDQ 1RWHWKDWDVXEVFULSWZDVDGGHGWRWKH SDUDPHWHUVWRGLVWLQJXLVKWKHPIURPWKRVHLQWKHRULJLQDOPRGHO 5HUXQQLQJ1//6IRUWKHPRGLILHGPRGHOWKHUHVXOWVLQ7DEOH DUH REWDLQHG $Q DQDO\VLV RI YDULDQFH GHOLYHUV D PRGHO )UDWLR YDOXH RI :LWK DQG GHJUHHVRIIUHHGRP IRU WKH PHDQVTXDUHG HUURUV RI WKH PRGHO DQG WKH HUURU UHVSHFWLYHO\ WKH PRGHO LV KLJKO\ VLJQLILFDQW 7DEOH([DPSOH,QWHUJDODFWLFYHORFLWLHV NPSHUVHFRQG
7DEOH([DPSOH(VWLPDWHVDQG &RQILGHQFH,QWHUYDOVIRUWKH500 3DUDPHWHUVGHULYHGIURP1//6 3DUDPHWHU $ %
(VWLPDWH
$V\PSWRWLF6(
&RQILG,QWHUYDO ^` ^`
CHAPTER 11 ESTIMATING THE ERROR DISTRIBUTION
189
Figure 11.3 displays the relative errors (in %). The mean absolute value of the errors is 0.3888%. Calculating the first four moments of W from the fitted (11.11), we find out that the mean and the variance are well preserved by the fitted equation: Calculated {µ,σ2}: {10.0, 0.007439} Sample {µ,σ2}: {9.970, 0.006929}. The sample skewness and kurtosis are not well preserved. This is to be expected given the small sample size and the well-known high MSE of direct sample estimates of skewness and kurtosis. Consider the KD solution. The authors concluded that the generalized beta distribution (GBD) may model well this dataset (therein, p. 208). The four-parameter d.f and CDF of GBD is given by (therein, p. 119) f(y) = (y-β1)β3(β1+β2y)β4 / [β(β3+1, β4+1)β2(β3+β4+1)], β1 \ β1+β2, F(y) =
∫
y
β1
f (u )du ,
(11.12) (11.13)
where β is the beta function: β(a,b) = Γ(a) Γ(b) / Γ(a+b). For the above data, KD obtain the fitted GLB parameters β1= 18675.3862, β2 = 4880.6411, β3 = -0.4789, β4 = -0.6045. Figure 11.4 displays the relative errors (in %). An error is defined as the difference between the quantile, yi, calculated from solving F(yi)= pi, and the actual i-th order statistic [pi is the CDF value for the i-th order statistic, calculated from (11.9)]. We realize that the fit is about the same as that obtained via (11.11) (refer to Figure 11.3). The mean absolute value of the relative errors is 0.28% (vs. 0.39% for the RMM model). 11.3. Moment-Based Estimation 11.3.1. Introduction Throughout this book reservation has been expressed about estimating a distribution's parameters via four-moment matching. Various sources had been quoted, including our own, where it was asserted that due to the
5(63216(02'(/,1*0(7+2'2/2*<
)LJXUH([DPSOH5HODWLYHHUURUV LQ RIWKHILWWHG YVDFWXDOORJRIWKH JDOD[LHVYHORFLWLHV 500PRGHO
)LJXUH ([DPSOH 5HODWLYH HUURUV LQ LQ WKH SUHGLFWHG ORJ TXDQWLOHV>EDVHV RQ WKH ILWWHG @ YV DFWXDO ORJ RI WKH LQWUDJDODFWLF YHORFLWLHV .' PRGHO EDVHG RQ *%'
&+$37(5(67,0$7,1*7+((5525',675,%87,21
ODUJH VDPSOLQJ HUURUV DVVRFLDWHG ZLWK VDPSOH HVWLPDWHV RI PRPHQWV RI KLJKGHJUHH WKLUG GHJUHH DQG KLJKHU WKH XVH RI WKHVH HVWLPDWHV LQ D ILWWLQJSURFHGXUHZDVQRWUHFRPPHQGHG $ VROXWLRQ WR WKLV SUREOHP KDV EHHQ DGGUHVVHG LQ WKLV ERRN VHYHUDO WLPHV,QHVVHQFHWKHVROXWLRQXVHVFRPSOHWHDQGSDUWLDOPRPHQWVRIORZ GHJUHH LQ WKH PDWFKLQJ SURFHGXUH ZKHUH SDUWLDO PRPHQWV UHIHU WR PRPHQWVFDOFXODWHGIRURQO\KDOIWKHGLVWULEXWLRQ7KLVPRPHQWPDWFKLQJ DSSURDFK HQVXUHV WKDW WKH DGYHUVH HIIHFW RI WKH ODUJH VWDQGDUG HUURUV DVVRFLDWHGZLWKHVWLPDWHVRIKLJKGHJUHHPRPHQWVLVDYHUWHG:HKDYH DOOXGHGWRWKLVDSSURDFKLQ&KDSWHUDQGZLOOGRWKDWDJDLQLQ&KDSWHUV DQG UHIHUDOVRWR6KRUH 7RDOORZHVWLPDWLRQRIWKHSDUDPHWHUVRIWKH500HUURUGLVWULEXWLRQ E\ PRPHQWPDWFKLQJ ZLWK ORZGHJUHH PRPHQWV WKH HUURU GLVWULEXWLRQ QHHGVWR KDYH D PDQDJHDEOH QXPEHU RI SDUDPHWHUV )RU H[DPSOHLI ZH ZLVKWRHVWLPDWHWKHSDUDPHWHUVXVLQJSDUWLDOPRPHQWVRIVHFRQGGHJUHH DW PRVW WKLV LPSOLHV WKDW DW PRVW IRXU SDUDPHWHUV QHHG WR EH HVWLPDWHG WZRIRUHDFKKDOIRIWKHGLVWULEXWLRQ $OWHUQDWLYHO\ D FRPELQDWLRQ RI DSSURDFKHV PD\ EH SXUVXHG )RU H[DPSOH ZH PD\ XVH D VDPSOH HVWLPDWH RI WKH UHVSRQVH PHGLDQ WR HVWLPDWH0$OVRZHPD\UHTXLUHPRPHQWSDULW\QRWRQO\IRUPRPHQWV RIWKHUHVSRQVHLQWKHRULJLQDOVFDOH < EXWDOVRIRUPRPHQWVLQWKHORJ WUDQVIRUPHGVFDOH : ,WKDVEHHQGHPRQVWUDWHG 6KRUH WKDWWKLV FRPELQDWLRQRIDSSURDFKHVUHVXOWVLQJRRGILWWLQJIRULQYHUVHQRUPDOL]LQJ WUDQVIRUPDWLRQV UHIHUWR&KDSWHUIRUGHWDLOV ,QGHHGVXFKPDUULDJHRIDSSURDFKHVLQDVLQJOHHVWLPDWLRQSURFHGXUH LV QRW DW DOO UDUH LQ WKH VWDWLVWLFV OLWHUDWXUH )RU H[DPSOH LQ ILWWLQJ WKH WKUHHSDUDPHWHU :HLEXOO GLVWULEXWLRQ &RKHQ :KLWWHQ DQG 'LQJ XVH ERWK WKH ILUVW WZR PRPHQWV PHDQ DQG YDULDQFH DQG WKH VPDOOHVW RUGHUVWDWLVWLF DSHUFHQWLOH WRHVWLPDWHWKH:HLEXOOSDUDPHWHUV /LNHZLVH WKH -RKQVRQ IDPLO\ LV RIWHQ ILWWHG EDVHG RQ PRPHQW PDWFKLQJ UHIHUWRDUHYLHZLQ6WDXUWDQG2UG VHFWLRQ SHUFHQWLOHPDWFKLQJ IRUH[DPSOH&KRX3RODQVN\DQG0DVRQ RU VRPHFRPELQDWLRQWKHUHRI ,QWKLVVHFWLRQHVWLPDWLRQSURFHGXUHVIRU F UHSDUDPHWHUL]HGLQ DUHGHYHORSHG(DUOLHUZHUHODWHGWRWKHDGYDQWDJHVRI IRU PRGHOLQJUDQGRPYDULDWLRQ 6HFWLRQ
5(63216(02'(/,1*0(7+2'2/2*<
7KHUHDUHIRXUSDUDPHWHUVLQ ^0$%'`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
<0 H[S^ Α >H[S %= ±@'=`
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≅ >$% $%' @ ^$% $$$ $%' $$ $%' $ $%''` &RQVLGHU DJDLQ ,W LV REYLRXV WKDW <0 U KDV WKH VDPH H[SUHVVLRQDV <0 ZLWK$DQG'UHSODFHGE\U$DQGU'UHVSHFWLYHO\
CHAPTER 11 ESTIMATING THE ERROR DISTRIBUTION
193
This implies that the approximate expression for E(Y/M) in (11.15) can also be used to approximate E[(Y/M)r], with the appropriate substitution for B and D, namely, E(Yr) ≅ (M)r{1 + (1/2)[rAB2 + r2(AB+D)2] + (3/24){rAB4(1 + 7rA + 6r2A2 + r3A3) + 4r2AB3D(1 + 3rA + r2A2) + 6r3AB2D2(1 + rA) + 4r4ABD3 + r4D4}}.
(11.16)
The reader may easily verify that (11.16) is a highly accurate approximation for the non-central moments of Y. To realize that, calculate for several values of the parameters the approximate moments, given by (11.16), and compare with the exact moments (to derive the latter, multiply Yr, as given by the RHS of (11.14), by the standard normal d.f and integrate). Since (11.16) provides an explicit expression for the r-th non-central moment, no numerical integration is needed during the search routine, and the parameters that deliver moment-matching may be easily identified (either via root-finding or by a minimization procedure). It was earlier explained that the moment-based estimation procedures involve matching of partial moments. Consider approximating the partial moments of Y in (11.14). We may wish to use the same approach that led to the derivation of an approximation to the complete moments (11.16). However, odd-order partial moments of the standard normal variable do not reduce to zero (as complete moments do). Therefore, into the expression derived from the Taylor expansion, we need to introduce for the odd-order power terms of Z partial moments of the standard normal variable. Denoting the r-th upper partial moment of Z by MU(Zr), namely, MU(Zr)=
∫
∞
0
z r φ ( z )dz ,
the first six partial moments are MU(Z) = (2π)-1/2 , MU(Z2) = 1/2 , MU(Z3) = (2/π)1/2 , MU(Z4) = 3/2, MU(Z5) = 4(2/π)1/2 , MU(Z6) = 15/2.
(11.17)
5(63216(02'(/,1*0(7+2'2/2*<
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
CHAPTER 11 ESTIMATING THE ERROR DISTRIBUTION
195
From (11.14), we have for the mean of W E(W) = log(M) + (Α)[exp(B2/2) – 1]. Introducing for "A" from (11.18) into (11.14) we obtain W = log(M)+[E(W)-log(M)][exp(BZ) – 1]/[exp(B2/2) – 1]+DZ
(11.18) (11.19)
This expression describes W in terms of log(M), E(W), and two unknown parameters, B and D. Denote by MU(Y) the response upper partial mean, namely, MU(Y) =
∫
∞
M
yf ( y )dy ,
(11.20)
where f(y) is the d.f. of Y. The lower partial mean, ML(Y), is equal to E(Y)-MU(Y). Estimates of both partial moments are obtained from the available sample of observations. M is set to be the sample median. From (11.19) we have Y = exp{log(M)+[E(W)-log(M)][exp(BZ)–1]/[exp(B2/2)–1]+DZ} (11.21) The upper partial moment of (11.21) is obtained from MU(Y) =
∫
∞
M
(M)exp{[E(W)-log(M)][exp(Bz)–1]/[exp(B2/2)–1] +Dz}φ(z)dz,
(11.22)
where φ(z) is the standard normal d.f. The lower partial moment is similarly defined. Equating the partial moments to the corresponding sample values, the parameters {B,D} may be obtained by any rootfinding routine. If one wishes to eliminate the numerical integration from the search routine, Taylor-based approximations may be developed, as explained earlier. Table 11.6 shows, for a sample of distributions, the parameters' values and the first four moments obtained with Procedure I. The first four moments are well preserved, even though only first-degree moments took part in the fitting procedure. Coupled with the low sampling errors associated with sample estimates of first-degree moments, this estimation procedure is expected to deliver good sampling accuracy.
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH3DUDPHWHUV YDOXHVDQGWKHILUVWIRXUPRPHQWVRI<REWDLQHGE\ILWWLQJ WRDVDPSOHRIGLVWULEXWLRQV([DFWPRPHQWVDUHXSSHUHQWULHVLQEROG0LGGOH HQWULHVUHIHUWR3URFHGXUH,DQGERWWRPHQWULHVWR3URFHGXUH,,0LVWKHPHGLDQ 'LVWULEXWLRQ 1RUPDO µ 0 σ *DPPD α β 0 *DPPD α β 0 :HLEXOO α β 0 :HLEXOO α β 0 :HLEXOO α β 0 :HLEXOO α β 0 5D\OHLJK σ 0 ([WUHPH9DOXH α β 0
$
%
'
0HDQ
9DU
6N
.X
3URFHGXUH,, 0DWFKLQJWKHPHDQDQGWKHYDULDQFHRI:DQG WKRVHRI< RUWKHSDUWLDOPHDQVRI< QRHVWLPDWHRIPHGLDQ 7KLV HVWLPDWLRQ SURFHGXUH XVHV VDPSOH VWDWLVWLFV RI ILUVW DQG VHFRQG GHJUHHRQO\$PDMRUVKRUWFRPLQJRI3URFHGXUH,LVWKDWWKHPHGLDQKDV WR EH HVWLPDWHG 6LQFH WKH PHGLDQ DSSHDUV DV D VFDOH IDFWRU LQ DQ\ VDPSOLQJ HUURU DVVRFLDWHG ZLWK WKH VDPSOH PHGLDQ ZLOO EH OLQHDUO\ WUDQVPLWWHG WR TXDQWLOH YDOXHV FDOFXODWHG IURP WKH HVWLPDWHG TXDQWLOH IXQFWLRQ$WUDGLWLRQDOVDPSOHHVWLPDWHRIWKHPHGDLQLVEDVHGRQDVLQJOH
&+$37(5(67,0$7,1*7+((5525',675,%87,21
REVHUYDWLRQ RUWZRDWPRVW 7KLVLPSOLHVKLJKVDPSOLQJLQDFFXUDF\,WLV WKHUHIRUH KLJKO\ GHVLUDEOH WKDW WKH QHHG WR VSHFLI\ WKH PHGLDQ EH HOLPLQDWHG LQ DQ\ HVWLPDWLRQ SURFHGXUH 7KLV LV D JXLGLQJ SULQFLSOH LQ GHYHORSLQJ3URFHGXUH,, )URP ZHREWDLQ
ORJ 0 µ : $>H[S % @
ZKHUH µ : LV WKH PHDQ RI : >HDUOLHU GHQRWHG E\ ( : @ ,QWURGXFLQJ EDFNLQWR ZHREWDLQ < H[S^ORJ 0 $>H[S %] ±@']`
H[S^µ : $>H[S %] H[S % @']`
)URP E\WDNLQJORJRIERWKVLGHV :µ : $^H[S %] (>H[S %= @`']
6TXDULQJERWKVLGHVDQGWDNLQJH[SHFWDWLRQVZHKDYH 9DU : $9DU>H[S %= @ $ % ' H[S % '9DU =
ZKHUH 9DU LV WKH YDULDQFH RSHUDWRU DQG % H[S % LV WKH H[SHFWHG YDOXHRI = H[S %= ,WLVHDV\WRVKRZWKDW 9DU>H[S %= @ H[S % ±H[S % ,QWURGXFLQJEDFNLQWR ZHREWDLQIRU' ! ' $ % H[S % ^$> % H[S % H[S % @9DU : `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
5(63216(02'(/,1*0(7+2'2/2*<
7ZRQXPHULFDOH[DPSOHV PRPHQWEDVHGHVWLPDWLRQ ([DPSOH%LUWKZHLJKWVRIWZLQV7KH,QGLDQD7ZLQ6WXG\ :H ZLOO XVH WKH7ZLQV 6WXG\ GDWD RI ([DPSOH LQ 6XEVHFWLRQ )RUWKLVH[DPSOH3URFHGXUHZLOOEHLPSOHPHQWHG 6XEVHFWLRQ 7KLVSURFHGXUHUHTXLUHVVDPSOHHVWLPDWHVRIWKHPHGLDQWKHPHDQRI: DQGWKHSDUWLDOPHDQVRI<)URPWKHGDWDLQ7DEOHZHREWDLQ 0HGLDQ < µ :
08 < 0/ <
1RWHWKDWWRGHULYHDQHVWLPDWHIRUWKHXSSHUSDUWLDOPRPHQWZHVXP DOOREVHUYDWLRQVODUJHUWKDQWKHPHGLDQDGGKDOIWKHYDOXHRIWKHPHGLDQ VLQFHWKHVDPSOHVL]HLVRGGQ DQGWKHQGLYLGHE\Q QRWQ 7KH HVWLPDWHIRUWKHORZHUSDUWLDOPHDQPD\EHVLPLODUO\GHULYHGRUVLPSO\ FDOFXODWHGE\VXEWUDFWLQJWKHXSSHUSDUWLDOPHDQIURPWKHFRPSOHWHPHDQ ,PSOHPHQWLQJ 3URFHGXUH , ZH REWDLQ WKH IROORZLQJ SDUDPHWHU HVWLPDWHV>UHIHUWR DQG @ % ' DQGIURP $ >( : ORJ 0 @>H[S % ±@ 7KH ILUVW IRXU PRPHQWV FDOFXODWHG QXPHULFDOO\ IURP WKH ILWWHG TXDQWLOHIXQFWLRQDUH ZLWKWKHVDPSOHHVWLPDWHVLQEUDFNHWV µ < 9DU < 6N .X :H UHDOL]H WKDW HYHQ WKRXJK VDPSOH HVWLPDWHV RI RQO\ ILUVWGHJUHH PRPHQWVWRRNSDUWLQWKHILWWLQJSURFHGXUHKLJKHUPRPHQWVDUHDOVRZHOO SUHVHUYHG 1RWH WKDW WKH YDOXH RI % LV FORVH WR WKDW REWDLQHG HDUOLHU E\ SHUFHQWLOHILWWLQJ DQGERWKDUHFORVHWRWKDWREWDLQHGLI LV ILWWHG WR D QRUPDO GLVWULEXWLRQ DERXW ,Q IDFWWKH FRQILGHQFH LQWHUYDOIRU% UHIHUWR7DEOH GRHVLQFOXGH LPSO\LQJWKDWWKH GDWDSUREDEO\RULJLQDWHGLQDQRUPDOGLVWULEXWLRQ
&+$37(5(67,0$7,1*7+((5525',675,%87,21
([DPSOH'LVWULEXWLRQRI,QWUD*DODFWLFYHORFLWLHV 7KHJDOD[\YHORFLW\REVHUYDWLRQVRI([DPSOH 6HFWLRQ DUHXVHG %RWK3URFHGXUH, 6HFWLRQ DQG3URFHGXUH,, 6HFWLRQ ZLOO EHGHPRQVWUDWHG 3URFHGXUH, 7KLVSURFHGXUHUHTXLUHVVDPSOHHVWLPDWHVRIWKHPHGLDQWKHPHDQRI: DQGWKHSDUWLDOPHDQVRI<)URPWKHGDWDLQ6HFWLRQZHREWDLQ 0HGLDQ < 08 < 0/ < µ : ,PSOHPHQWLQJ 3URFHGXUH , ZH KDYH WKH IROORZLQJ SDUDPHWHU HVWLPDWHV % '
$ >( : ORJ 0 @>H[S % @
/LNHLQWKHHDUOLHUDQDO\VLVRIWKLVH[DPSOH 6HFWLRQ LWLVHDV\ WRUHDOL]HWKDWWKHSDUDPHWHUV $% DQG'DUHYHU\FORVHLQWKHLUDEVROXWH YDOXHVVRWKHIROORZLQJFRUUHFWHGPRGHOZDVWULHG < 0 H[S^$^ % >H[S %= @=`` ZKHUH 0 LV WKH VDPSOH PHGLDQ 7KLV PRGHO QRZ KDV RQO\ WZR SDUDPHWHUV EHVLGHVWKHPHGLDQ VRLWPLJKWEHDJRRGLGHDWRVZLWFKWR PDWFKLQJWKHILUVWWZRPRPHQWVQDPHO\WKHPHDQDQGWKHYDULDQFH )URP XVLQJD7D\ORUVHULHVH[SDQVLRQDSSUR[LPDWLRQ>VLPLODU WR WKH GHULYDWLRQ RI @ ZH REWDLQ WKH DSSUR[LPDWH H[SOLFLW H[SUHVVLRQIRUWKHUWKFRPSOHWHPRPHQWRI< ( U%$ %U$ U%$ U%$U$
U$ %U$%U$U$ U%$U$ @`
8VLQJ WKLV DSSUR[LPDWH KLJKO\ DFFXUDWH H[SUHVVLRQ ZH REWDLQ E\ PDWFKLQJWKHPHDQVDQGWKHYDULDQFHVWKHSDUDPHWHUV HVWLPDWHV $ %
5(63216(02'(/,1*0(7+2'2/2*<
7KH ILUVW IRXU PRPHQWV RI WKH ILWWHG TXDQWLOH IXQFWLRQ DUH ZLWK WKH VDPSOHPRPHQWVLQEUDFNHWV 0 µ <
9DU < > @
6N < .X <
*LYHQ WKH VPDOO VDPSOH DQG WKH UHODWLYHO\ KLJK VDPSOLQJ HUURUV DVVRFLDWHGZLWKVDPSOHHVWLPDWHVRIVNHZQHVVDQGNXUWRVLVWKHPRPHQWV FDOFXODWHG IURP WKH ILWWHG DUH FORVH WR WKH VDPSOH YDOXHV 7KH UHDGHU VKRXOG DJDLQ EH UHPLQGHG WKDW RQO\ ILUVW DQG VHFRQGGHJUHH PRPHQWVSDUWLFLSDWHGLQWKHILWWLQJSURFHGXUH 3URFHGXUH,, 7KLV SURFHGXUH UHTXLUHV PDWFKLQJ WKH PHDQ DQG WKH YDULDQFH RI : ORJ < DQG WKH SDUWLDO PHDQV RI < 1R (VWLPDWH RI WKH PHGLDQ LV QHHGHG7KHVDPSOHHVWLPDWHVDUH
µ : 9DU :
6N : .X :
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
&+$37(5(67,0$7,1*7+((5525',675,%87,21
JLYHQ E\ )XUWKHUPRUH WKH HDUOLHU VROXWLRQ ZKHUH WKHVH SDUDPHWHUV ZHUH GHWHUPLQHG E\ PDWFKLQJ RI WKH UHVSRQVH PHDQV DQG YDULDQFHVSURYLGHVWKHEHVWHVWLPDWHVIRUWKHXQNQRZQSDUDPHWHUV :H FRQFOXGH ZLWK D FRPPHQW UHJDUGLQJ HPSLULFDO PRGHOLQJ DV WKH ODWWHU PDWHULDOL]HG IRU WKH JDOD[\ V YHORFLW\ GDWD ([DPSOH 6LQFH % FRQVLVWHQWO\WXUQHGRXWWREHFORVHWR]HURLWZDVUHDOL]HGWKDWDGLIIHUHQW PRGHOZDVVXJJHVWHGE\WKHGDWDDQGZKDWHYHUVROXWLRQSURFHGXUHZDV LPSOHPHQWHGWKHUHVXOWVDOZD\VSRLQWHGWRWKHVDPHPRGHO $JRRGOHVVRQPD\EHOHDUQHGIURPWKLVH[DPSOH,IWKHILWDFKLHYHG LV QRW VDWLVIDFWRU\ WDNH D JRRG ORRN DW WKH YDOXHV RI WKH SDUDPHWHUV HVWLPDWHV,QPRVWFDVHVWKH\WHOODVWRU\DQGPD\GLUHFW\RXWRWKHEHVW ZD\ WR PRGLI\ \RXU PRGHO ,Q RXU H[DPSOH WKH IDFW WKDW % ZDV FRQVLVWHQWO\ FORVH WR ]HUR LQ DOO VROXWLRQ SURFHGXUHV DWWHPSWHG LPPHGLDWHO\ VXJJHVWHG WKDW LQVWHDG RI PRGHOLQJ WKLV SDUDPHWHU E\ WKH WHUP H[S %= SHUKDSV WKH WHUP >H[S %] @% ZRXOG SURYLGH EHWWHU JRRGQHVVRIILW /LNHZLVHWKH SUR[LPLW\ LQ WKH DEVROXWH YDOXHV RI $% DQG ' LPPHGLDWHO\ VXJJHVWHG WKDW SHUKDSV RQH SDUDPHWHU LV VXSHUIOXRXV D IDFW WKDW ZRXOG SUREDEO\ PDQLIHVW LWVHOI LQ KLJK FRUUHODWLRQVLIHVWLPDWHV FRUUHODWLRQVZHUHFDOFXODWHG /RRNLQJ FDUHIXOO\ DW WKH UHVXOWV RI WKH VWDWLVWLFDO DQDO\VLV OHDUQLQJ IURP WKHP DQG WKHQ PRGLI\LQJ WKH PRGHOLQ DQ LWHUDWLYHIDVKLRQXQWLO WKH EHVW ILW LV DFKLHYHG LV DOZD\V D JRRG WLPHKRQRUHG SUDFWLFH IRU HPSLULFDOPRGHOLQJ 5HIHUHQFHV >@&KRX<03RODQVN\$00DVRQ5/ 7UDQVIRUPLQJQRQQRUPDOGDWDWR QRUPDOLW\LQVWDWLVWLFDOSURFHVVFRQWURO-RXUQDORI4XDOLW\7HFKQRORJ\ >@&RKHQ$&:KLWWHQ%-'LQJ< 0RGLILHGPRPHQWHVWLPDWLRQIRUWKH WKUHHSDUDPHWHU:HLEXOOGLVWULEXWLRQ-RXUQDORI4XDOLW\7HFKQRORJ\ >@'RGVRQ% :HLEXOO$QDO\VLV$643UHVV0LOZDXNHH:, >@ .DULDQ = $ 'XGHZLF] ( - )LWWLQJ 6WDWLVWLFDO 'LVWULEXWLRQV 7KH *HQHUDOL]HG /DPEGD 'LVWULEXWLRQ DQG *HQHUDOL]HG %RRWVWUDS 0HWKRGV &5& 3UHVV%RFD5DWRQ)ORULGD86$
5(63216(02'(/,1*0(7+2'2/2*<
>@ .DULDQ = $ 'XGHZLF] ( - &RPSDULVRQ RI */' ILWWLQJ PHWKRGV 6XSHULRULW\ RI SHUFHQWLOH ILWV WR PRPHQWV LQ / QRUP -RXUQDO RI WKH ,UDQLDQ 6WDWLVWLFDO6RFLHW\ >@ 2HJHUOH : 5 +LOO - 0 )LWFKHWW 0 - 2EVHUYDWLRQV RI KLJK GLVSHUVLRQ FOXVWHUV RI JDOD[LHV &RQVWUDLQWV RQ FROG GDUN PDWWHU 7KH $VWURQRPLFDO -RXUQDO S >@6WXDUW$2UG. .HQGDOO¶V$GYDQFHG7KHRU\RI6WDWLVWLFV9 'LVWULEXWLRQ 7KHRU\&KDUOHV*ULIILQ &RPSDQ\/WG/RQGRQ >@ 6KRUH + *HQHUDO FRQWURO FKDUWV IRU YDULDEOHV ,QWHUQDWLRQDO -RXUQDO RI 3URGXFWLRQ5HVHDUFK > @ 6KRUH + 1RQQRUPDO SRSXODWLRQV LQ TXDOLW\ DSSOLFDWLRQV $ UHYLVLWHG SHUVSHFWLYH4XDOLW\DQG5HOLDELOLW\(QJLQHHULQJ,QWHUQDWLRQDO
&KDSWHU
6SHFLDO&DVHVRIWKH5000RGHO
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
5(63216(02'(/,1*0(7+2'2/2*<
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
< 0 H[S^ αλ > ηε1 λ@ε@`
D
< 0 H[S^ αλ > ηε1 λ@` ε2)
E
< 0 H[S^ αλ >ηλH[S λε1 η @ε2`
F
< 0 H[S^ αλ >ηλH[S λε1 η @` ε2
G
1RWH WKDW RFFDVLRQDOO\ 0 LQ WKHVH PRGHOV ZLOO EH LQWURGXFHG LQ WKH H[SRQHQWLDODVµ2 [=ORJ(Μ)]. /LWHUDWXUH UHODWHG WR WKH PRGHOV RI WKLV VHFWLRQ KDV EHHQ JLYHQ LQ &KDSWHU$FFRUGLQJO\RQO\QHZUHIHUHQFHVWKDWDUHQRWPRGHOUHODWHG DUHSURYLGHGLQWKHUHIHUHQFHVOLVWDWWKHHQGRIWKLVFKDSWHU &KHPLVWU\DQGFKHPLFDOHQJLQHHULQJ :H VWDUW ZLWK WHPSHUDWXUH 7 GHSHQGHQFH RI YDSRU SUHVVXUH 3 5HIHUULQJWR 6HFWLRQ
ORJ 3 $% µ7ε& ε2%
ZKHUH7 µ7εDQGµ7LVWKHH[SHFWHGYDOXH,WLVHDVLO\UHFRJQL]HGWKDW WKLVHTXDWLRQLVGHULYDEOHIURPWKH500PRGHO D ZLWKSDUDPHWHUV η µ7&α %λ µ $% D (TXDWLRQ LVDSRO\QRPLDODQGZLOOEHDGGUHVVHGVHSDUDWHO\ )RUVXUIDFHWHQVLRQ6>(T @UHZULWWHQKHUHZLWKWKHHUURUVZH KDYH
6 $> µ7ε1)7F@& ε2
&+$37(563(&,$/&$6(62)500
,W LV HDV\ WR YHULI\ WKDW FDQ EH GHULYHG IURP E ZLWK SDUDPHWHUV η µ77Fα &λ 0= $ D 5HODWLQJ QH[W WR ZH UHDOL]H WKDW WKH XSSHU WZR HTXDWLRQV DUH VSHFLDO FDVHV RI WKH 500 PRGHO H[SUHVVLQJ WKH UHVSRQVH DV H[SRQHQWLDOSRZHU DQG SRZHU UHODWLRQVKLSV UHVSHFWLYHO\ 7KH WKLUG HTXDWLRQLVDSRO\QRPLDOZKLFKZLOOEHDGGUHVVHGVKRUWO\ )RUVROLGGHQVLW\'DXEHUW QRWHV UHIHUWR&KDSWHU WKDW«DOLQHDU HTXDWLRQ LV QRUPDOO\ DGHTXDWH « ,Q D IHZ FDVHV « D VLPSOH TXDGUDWLF SRO\QRPLDO ZDV UHFRPPHQGHG )RU PRVW VLWXDWLRQV XVH RI D VLQJOH H[SRQHQWLDO YDOXH RYHU WKH HQWLUH WHPSHUDWXUH UDQJH LV VXIILFLHQW $OO WKHVH FDVHV DUH VWHSV RQ WKH /DGGHU VSHFLDO FDVHV RI WKH 500 PRGHO )RU OLTXLG GHQVLW\ ρ/ 5DFNHWW V HTXDWLRQ PD\ EH UHH[SUHVVHG ZLWKWKHHUURUVE\
ORJ ρ/ $%> µ7ε1 &@'ε2
ZKHUH $ % & DQG ' DUH SDUDPHWHUV WKDW QHHG WR EH GHWHUPLQHG ,W LV HDV\WRVHHWKDWWKLVHTXDWLRQLVGHULYDEOHIURP D ZLWK η µ7&α %'λ 'µ2 Α+Β D )RUVROLGKHDWFDSDFLW\'DXEHUWUHFRPPHQGVDVLPSOHSRO\QRPLDOLQ WHPSHUDWXUHZKHUHPRVWGDWDFDQEHILWWHGZLWKDOLQHDUHTXDWLRQZLWKD TXDGUDWLFQHFHVVDU\IRUDIHZV\VWHPV%RWKWKHOLQHDUDQGWKHTXDGUDWLF DUHVSHFLDOFDVHVRI500 )RUOLTXLGKHDWFDSDFLW\'DXEHUWUHFRPPHQGVXVLQJDSRO\QRPLDOLQW >(TXDWLRQ @ZKHUHW 7U3RO\QRPLDOVZLOOEHDGGUHVVHGVKRUWO\ )RULGHDO JDV KHDW FDSDFLW\ &3R 'DXEHUWUHFRPPHQGV WKH XVH RI DQ H[SRQHQWLDO >(TXDWLRQ @ 5HZULWLQJ LW ZLWK WKH UDQGRP HUURUV ZH REWDLQ
&3R $^%H[S>& µ7ε1)'@` ε
,WLVHDV\WRVHHWKDWWKLVLV E FRUUHFWHGIRUORFDWLRQZLWK η µ7α ' & λ 'µ2 ORJ % ±&
D
)RUWHPSHUDWXUHGHSHQGHQF\RIOLTXLGYLVFRVLW\'DXEHUW UHFRPPHQGV DSRO\QRPLDO>(TXDWLRQ @WREHDGGUHVVHGVKRUWO\
5(63216(02'(/,1*0(7+2'2/2*<
)RU WKHUPDO FRQGXFWLYLW\ 'DXEHUW H[DPLQHV ERWK OLTXLG DQG ORZ SUHVVXUHYDSRUWKHUPDOFRQGXFWLYLW\DOORIZKLFKDUHHLWKHUSRO\QRPLDOV LQ7RULQW 7U 77F 7KH $UUKHQLXV HTXDWLRQ KDV WKH VDPH VWUXFWXUH DV ZLWK ORFDWLRQ SDUDPHWHU HTXDO ]HUR DQG WKHUHIRUH LV HDVLO\ UHFRJQL]HG DV D VSHFLDOFDVHRIWKH500PRGHO :H QRZ FRQVLGHU SRO\QRPLDO PRGHOV 3RO\QRPLDOV DUH D OLQHDU FRPELQDWLRQ RI YDULRXV SRZHUV RI WKH OLQHDU SUHGLFWRU η DQG WKHUHIRUH FRQVWLWXWH D OLQHDU FRPELQDWLRQ RI YDULRXV SRZHU WHUPV GHULYDEOHIURPWKH500PRGHOZLWKλ=6LQFHDQLPSOLFLWREMHFWLYHRI XVLQJSRO\QRPLDOVLVRIWHQWRFDSWXUHYDULRXVGHJUHHVRIFRQYH[LW\RQH FDQUHDVRQDEO\H[SHFWWKH500PRGHOWRSURYLGHYLDEOHVXEVWLWXWHWRWKH XVHRISRO\QRPLDOV $GHPRQVWUDWLRQRIWKLVLVJLYHQE\YLVFRVLW\PRGHOLQJGHOLYHUHGLQ 'DXEHUW E\ WZR H[SUHVVLRQV >(TXDWLRQV DQG WKHUHLQ@ DQGSUHVHQWHGKHUHLQDVRPHZKDWGLIIHUHQWIRUP
%
µ/ H[S>$%7&ORJ 7 '7(@
µ/ H[S $) &
ZKHUH) 7 'DXEHUWVWDWHVWKDW DQG DUHQHDUO\HTXLYDOHQW LQWKHGHJUHHRIDFFXUDF\WKH\GHOLYHUUHIHUWR6KRUH %UDXQHU DQG 6KDFKDP DQG 6KRUH @ 7KH SRLQW LV IXUWKHU GLVFXVVHGLQ&KDSWHU 7KHLVVXHRIZKHWKHUDSRO\QRPLDOVKRXOGEHXVHGKDVWREHFDUHIXOO\ EDODQFHGDJDLQVWWKHDOWHUQDWLYHRIIHUHGE\WKH500DSSURDFK
&+$37(563(&,$/&$6(62)500
3K\VLFV ,QWURGXFLQJ NLQHWLF HQHUJ\ (N DV D IXQFWLRQ RI YHORFLW\ 9 ZLWK WKH HUURUVDGGHGZHREWDLQIURP (N 0 µ9ε1 ε2
ZLWK9 µ9ε16LQFHWKLVLVDSRZHUUHODWLRQVKLSLWLVDVSHFLDOFDVHRI WKH500PRGHO )RU1HZWRQ VODZRIFRROLQJ ZHREWDLQZLWKWKHHUURUVDGGHG ∆7 ∆7H[S>D µW+ε1) + ε2@
ZKHUHPHDVXUHGWLPHLVJLYHQE\W µW+ε1.7KLVLVDQ500PRGHOZLWK SDUDPHWHUV λ α D )RU WKH ODZ ZKLFK ELQGV WRJHWKHU VDWXUDWLRQ YDSRU SUHVVXUH 3YDSRUZLWKWHPSHUDWXUH7 µ7ε1ZHKDYHZLWKWKHHUURUVDGGHG 3YDSRU∼H[S^(%>κ% µ7ε1)@ε2},
ZKHUH κ% LV WKH %ROW]PDQQ IDFWRU 7 LV RQ D .HOYLQ VFDOH DQG (E LV D SDUWLFOH ELQGLQJ HQHUJ\ 7KLV PRGHO PD\ EH GHULYHG IURP 500 ZLWK SDUDPHWHUVλ α (Eκ% 7KH%DURPHWULFIRUPXODLV>(T ZLWKWKHHUURUVDGGHG@ ρ µ. ρ H[S^ 0J5 µ.ε1 ε2`
7KLVPRGHOLV D ZLWKSDUDPHWHUV λ , α 0J5 )URP (LQVWHLQ V 6SHFLDO7KHRU\ RI 5HODWLYLW\ WKH IRUPXOD IRU WRWDOHQHUJ\FDUULHGE\DPRYLQJREMHFWLV ZLWKRXWHUURUV ( / 0& /
ZLWK/ Y& 7KLVSRZHUUHODWLRQVKLSFDQEHGHULYHGIURP500ZLWK SDUDPHWHUVη /λ α 3ODQFN VUDGLDWLRQODZLV>(T ZLWKWKHHUURUε1, DGGHG@
6 µ7 πFKλ0 ^H[S> KFλ0N µ7ε1)@`
ZKHUH 6 LV VSHFWUDO UDGLDQF\ λ0 LV WKH ZDYHOHQJWK DW WHPSHUDWXUH 7 µ7ε1DQGKLVWKH3ODQFNFRQVWDQW7KHUHVSRQVH πFKλ0 6 µ7 D OLQHDU WUDQVIRUPDWLRQ RI 6 LV DQ H[SRQHQWLDOSRZHU UHODWLRQVKLS GHULYDEOHIURP500ZLWKSDUDPHWHUVλ α KFλ0N 7KH HQHUJ\ OHYHOV RI WKH VWDWLRQDU\ VWDWHV RI WKH K\GURJHQ DWRP DUH >(T UHLQWURGXFHGKHUHIRUFRQYHQLHQFH@ ( Q H9QQ
5(63216(02'(/,1*0(7+2'2/2*<
ZKHUH Q LV WKH TXDQWXP QXPEHU DQG WKH QXPHULFDO FRHIILFLHQW LV WKH HOHFWURQ JURXQGVWDWH HQHUJ\ PHDVXUHG LQ HOHFWURQYROWV H9 7KLV PRGHO FDQ EH GHULYHG IURP 500 ZLWK SDUDPHWHUV η Q λ α 6LQFHWKLVLVDWKHRUHWLFDOPRGHODQGQLVDQLQWHJHUQRHUURUVDUHDGGHG 5DGLRDFWLYH GHFD\ UDWH 5 LV JLYHQ E\ >(T ZLWK WKH HUURUV DGGHG@
5 µW ^5H[S>N µWε1 @` ε2),
ZLWKWLPHW µWε17KLVLVDQ500PRGHOZLWKSDUDPHWHUV λ α N 7KHSRZHURXWSXWDWDQXFOHDUUHDFWRUDWWLPHW3 W LVJLYHQE\>(T ZLWKWKHHUURUVDGGHG@ 3 µW 3H[S> NWJHQ µWε1) + ε2] ZKHUHW µWε17KLVLV500ZLWKSDUDPHWHUV λ α NWJHQ (OHFWULFDOHQJLQHHULQJ &RXORPE VODZLV>(T ZLWKWKHHUURUVDGGHG@
) µU ^N TT µUε1)` ε2)
ZKHUH U µUε1 LV WKH GLVWDQFH 7KLV PRGHO LV D SRZHU ODZ DQ 500 PRGHOZLWKSDUDPHWHUV λ α )RUDFKDUJLQJFDSDFLWRUFRQQHFWHGWRDFKDUJLQJVRXUFHKDYLQJHPI ( WKH YROWDJH DFURVV WKH FDSDFLWRU DW WLPH W DV PHDVXUHG E\ D UHVLVWRU FRQQHFWHGLQSDUDOOHOLV>(T ZLWKWKHHUURUVDGGHG@
95 µW (H[S> µWε1)5&ε2]
ZLWKW µWε1. 6LPLODUO\DGLVFKDUJLQJFDSDFLWRUZLWKDQLQLWLDO IXOO FKDUJHRIT ZLOOKDYHDWWLPHWDFKDUJHRI>(T ZLWKWKHHUURUVDGGHG@
T µW TH[S> µWε1)5&ε2@
%RWK DQG DUH500PRGHOVJRYHUQHGE\DQH[SRQHQWLDO UHODWLRQVKLS )RU DQ HOHFWULFLW\ FXUUHQW RI LQWHQVLW\ , WKDW UXQV WKURXJK D ORRS RI UDGLXV5 DUHD6 WKHPDJQHWLFILHOGLV>(T ZLWKWKHHUURUVDGGHG@
% µG > µπ) ,6 µGε1)@ ε2)
> µ) ,5 µGε1)@ ε2
&+$37(563(&,$/&$6(62)500
7KLV500PRGHOKDVSDUDPHWHUVλ α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
= γ + δORJ>< < @<
DQGWKH68IDPLO\
= γ + δ VLQK< γ + δ ORJ>< < @<
ZKHUH < ;ξ)/λ, ; LV WKH RULJLQDO UDQGRP YDULDEOH WKDW ZH ZLVK WR PRGHOYLDDPHPEHURIWKH-RKQVRQIDPLO\DQGξ DQGλ ! DUHORFDWLRQ DQGVFDOHSDUDPHWHUVUHVSHFWLYHO\ &RQVLGHUILUVWWKH6%IDPLO\,QWURGXFLQJ8 =γ)/δZHREWDLQIURP
RU
5(63216(02'(/,1*0(7+2'2/2*<
8 ORJ>< < @
< H[S>ORJ H8 8@
,I LQ WKH 500 PRGHO ZLWK η >UHODWH WR D @ ZH DVVXPH WKDW ε H[S 8 ε2 8ZHREWDLQ
< H[S^ αλ > 1H8 λ±@µ 8`
7KLV LV HTXLYDOHQW WR ZLWK λ α µ 1RWH WKDW LQ GHULYLQJ ZHKDYHPRGLILHGWKHDVVXPSWLRQDERXWWKHGLVWULEXWLRQ RIWKHHUURUε DVVXPLQJORJQRUPDOLW\LQVWHDGRIQRUPDOLW\ *LYHQWKH DSSUR[LPDWHHTXLYDOHQFHRIWKHVHWZRGLVWULEXWLRQVDVUHSUHVHQWDWLRQVRI WKHHUURUV GLVWULEXWLRQ ZHKDYHDOOXGHGWRWKLVHDUOLHULQ&KDSWHU WKH PRGLILHGDVVXPSWLRQGRHVQRWYLRODWHWKHEDVLFHTXLYDOHQFHEHWZHHQWKH 500HUURUGLVWULEXWLRQDQGWKH6%IDPLO\ )RUWKH68V\VWHPZHKDYHIURP < >H[S 8 ±H[S 8 @ >H[S 8 @H[S 8
RU
< H[S^ORJ>H[S 8 @ORJ 8`
,WLVHDVLO\UHFRJQL]HGWKDW LVDQ500PRGHO)URP D LQWURGXFLQJη ZHREWDLQ
< H[S^ αλ > 1ε1 λ@µ2ε@`
7KLVLV ZLWKα=1, λ , µ2= ORJ DQGHUURUV ε H[S 8 ε2= −8 1RWH WKDWDJDLQ ε LV ORJQRUPDOO\ GLVWULEXWHG DQG WKXV GHYLDWH IURP RXU LQLWLDO DVVXPSWLRQ &KDSWHU WKDW ERWK HUURUV DUH QRUPDOO\ GLVWULEXWHG +RZHYHU IRU 8 H[S 8 ≈ 8 DQG WKH LQLWLDO DVVXPSWLRQDERXWWKHDSSUR[LPDWHQRUPDOLW\RIERWKHUURUVLVUHVWRUHGIRU WKLVFDVH $WKLUGWUDQVIRUPDWLRQWKDWEHORQJVWRWKH-RKQVRQIDPLO\UHVXOWVLQ< KDYLQJDORJQRUPDOGLVWULEXWLRQDJDLQDQ500PRGHO α 7XNH\JDQGKV\VWHPVRIGLVWULEXWLRQV ,Q +RDJOLQ 0RVWHOOHU DQG 7XNH\ +RDJOLQ LQWURGXFHV DQ DSSURDFK RULJLQDOO\ LQWURGXFHG E\ 7XNH\ WR PRGHO UDQGRP YDULDEOHV LQ WHUPV RI WUDQVIRUPDWLRQV RI WKH XQLW QRUPDO YDULDEOH 7KLV
&+$37(563(&,$/&$6(62)500
DSSURDFKUHVXOWVLQ7XNH\ VJDQGKV\VWHPVZKLFKZHUHVXJJHVWHGWR PRGHO VNHZHG YDULDEOHV DQG HORQJDWHG V\PPHWULF YDULDEOHV UHVSHFWLYHO\&RPELQLQJWKHWZRV\VWHPVWRUHSUHVHQWERWKVNHZQHVVDQG HORQJDWLRQUHVXOWLQWKHJKV\VWHP 7KHWKUHHV\VWHPVRIGLVWULEXWLRQVDUHUHVSHFWLYHO\ UHIHUWR+RDJOLQ HWDO &KDSWHU <J = >H[S J= @JJ
<JK = J >H[S J= @H[S K= J
ZKHUH=LVDXQLWQRUPDOYDULDEOH 7KHVHV\VWHPVPD\EHGHULYHGDVVSHFLDOFDVHVIURPWKH500HUURU GLVWULEXWLRQ 5HIHUULQJ ILUVW WR VLQFH <J = LV D ORJQRUPDO YDULDEOH DGMXVWHG IRU ORFDWLRQ DQG VFDOH LQ D SDUWLFXODU PDQQHU LW LV D VSHFLDOFDVHRIWKH500HUURUGLVWULEXWLRQ 7RREWDLQ ZHKDYHIURPWKH500HUURUGLVWULEXWLRQ>UHIHUWR E ZLWKα=Kλ ORJ 0 K@
< H[S^ K ε1 2` ε2).
$VVXPH WKDW LQ ε1= ε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
; ;± ORJ> ρ ρ @
5(63216(02'(/,1*0(7+2'2/2*<
ZKHUH ρ LV WKH SRSXODWLRQ FRUUHODWLRQ FRQYHUJHV WR QRUPDOLW\ PRUH UDSLGO\WKHQULWVHOI)XUWKHUPRUHWR2 Q WKHPHDQRI; LV()ρ Q DQG WKH YDULDQFH LV Q ZKHUH Q LV WKH VDPSOH VL]H 7KH VNHZQHVV PHDVXUH 6N DQG WKH NXUWRVLV PHDVXUH .X DUH 62 9 S DQG .HQGDOODQG6WXDUW S
6N ρ3 Q Q .X Q 2 Q
7KLVLPSOLHVWKDWDOWKRXJK)LVKHU VWUDQVIRUPDWLRQLQWHQGVWRVWDELOL]H WKH YDULDQFH LW DOVR WHQGV WR QRUPDOL]H VLQFH WKH VNHZQHVV PHDVXUHV DSSURDFK WKDW RI WKH QRUPDO GLVWULEXWLRQ PRUH UDSLGO\ WKDQ U >2 Q FRPSDUHGWR2 Q UHVSHFWLYHO\@ $VVXPHWKDW
= Q >; ρ Q @ γ + δ;
LVDSSUR[LPDWHO\DXQLWQRUPDOYDULDEOH5HZULWLQJDV8 =γ δ= ;ZH REWDLQIURP U >H[S 8 @>H[S 8 @ ZKHUH8 FRUUHVSRQGVWRU ,QWURGXFLQJLQWR E ε1=H[S 8 ε2= −ε1, ZHREWDLQWKHUHRI
< 0 H[S^ αλ > H[S 8 λ@`>H[S 8 @
7KLVLVHTXLYDOHQWWR ZLWK0 λ=, α $JDLQRQO\VOLJKW FKDQJHV LQ WKH DVVXPSWLRQV DERXW WKH GLVWULEXWLRQV RI WKH HUURUV ZHUH QHHGHGWRREWDLQWKH)LVKHUWUDQVIRUPDWLRQIURPWKH500HUURUTXDQWLOH IXQFWLRQ +DOGDQHSRZHUWUDQVIRUPDWLRQDQG:LOVRQ+LOIHUW\ DSSUR[LPDWLRQWRχ2 :LOVRQDQG+LOIHUW\ VXJJHVWHGDFXEHURRWWUDQVIRUPDWLRQRIWKH χ2 YDULDEOH ZLWKνGHJUHHVRIIUHHGRP LQRUGHUWRQRUPDOL]HLW
= ^ χν) ν ` ν
ZKHUH = LV DSSUR[LPDWHO\ D XQLW QRUPDO YDULDEOH +DOGDQH JHQHUDOL]HG WKLV UHVXOW DQG GHULYHG WZR WUDQVIRUPDWLRQV IRU D UDQGRP YDULDEOH;ZKLFKKDVDOOLWVFXPXODQWVRIRUGHUQZKHUHQLVWKHVDPSOH VL]H IRUH[DPSOHWKHELQRPLDOKDVWKLVSURSHUW\ /HWκMEHWKHMWKFXPXODQWRI;'HILQLQJ
&+$37(563(&,$/&$6(62)500
< ;κ K > ;κ1)/κ1@K
+DOGDQH KDV VKRZQ WKDW LI ZH SXW K κκ3/ κ < WKHQ KDV VNHZQHVV RI RUGHU Q DV FRPSDUHG WR Q IRU ; 0RUH UDSLG FRQYHUJHQFHWRQRUPDOLW\LVDFKLHYHG)XUWKHUPRUHE\DGGLQJDFRQVWDQW WR;WRPDNHLWVPHDQHTXDOWRJDQGFKRRVLQJ J κκ κ κκ K κ κκ κ κκ WKHUHGHILQHGYDULDEOH < > ;κ1)/J@K
KDV VNHZQHVV RI RUGHU Q DV EHIRUH DQG NXUWRVLV RI RUGHU Q FRPSDUHGWRQIRU; )LQGGHWDLOVLQ.HQGDOODQG6WXDUW S /HWXVUHZULWHWKHSRZHUWUDQVIRUPHGYDULDEOHDV η + δ= ξ + β; K ZKHUH = LV VWDQGDUG QRUPDO DQG DOO WKH RWKHU WHUPV DUH UHDOYDOXHG SDUDPHWHUV7KHQ+DOGDQH VWUDQVIRUPDWLRQLPSOLHV
; ξ/β) β η + δ= K
7KLV LV D SRZHU WUDQVIRUPDWLRQ RI D QRUPDO YDULDEOH D OLQHDU WUDQVIRUPDWLRQRIDVSHFLDOFDVHRI500 λ α K %R[&R[QRUPDOL]LQJWUDQVIRUPDWLRQ 7KLVZHOONQRZQWUDQVIRUPDWLRQ %R[DQG&R[ LVZLGHO\XVHGLQ HPSLULFDO UHODWLRQDO PRGHOLQJ ZKHQ WKH UHVSRQVH LV QRQQRUPDO 7KH WUDQVIRUPDWLRQ LV FRPPRQO\ DSSOLHG WR WKH UHVSRQVH VR WKDW WKH QRUPDO VFHQDULRLVUHYRNHG DQG OLQHDU UHJUHVVLRQ DQDO\VLVFDQ EHDSSOLHG WRWKH WUDQVIRUPHG GDWD ,Q WKLV VXEVHFWLRQ ZH DGGUHVV WKH %R[&R[ %& QRUPDOL]LQJWUDQVIRUPDWLRQDVDQHPSLULFDOPRGHOIRUUDQGRPYDULDWLRQ 'HQRWHE\< λ WKH%&WUDQVIRUPHGYDULDEOH
< λ
< ORJ <
1RWHWKDWWKHSDUDPHWHURIWKLVWUDQVIRUPDWLRQLVλ WRGLVWLQJXLVKLWIURP WKH500SDUDPHWHU λ. 5HZULWLQJ < λ η δ= ZKHUH = LV D XQLW QRUPDO YDULDEOH ZH REWDLQIURP
5(63216(02'(/,1*0(7+2'2/2*<
< >λ η δ= @λ λ η λ δ=)1/λ
7KLVLPSOLHVWKDWWKH%&WUDQVIRUPDWLRQWDFLWO\DVVXPHWKDW<FDQEH H[SUHVVHGDVDSRZHUWUDQVIRUPDWLRQRIDQRUPDOYDULDEOH 7KH LQYHUVH %& WUDQVIRUPDWLRQ LV GHULYDEOH IURP WKH 500 PRGHO D ZLWKSDUDPHWHUV η λ η σε1 λ δ, λ α λ µ σε2 )RUλ WKHLQYHUVH%&WUDQVIRUPDWLRQUHVXOWVLQDORJQRUPDOYDULDEOH DOVRDVSHFLDOFDVHRIWKH500PRGHO &DXFK\GLVWULEXWLRQ /HW8DQG9EHLQGHSHQGHQWVWDQGDUGQRUPDOYDULDEOHV7KHUDWLR < 89 KDVDVWDQGDUG&DXFK\GLVWULEXWLRQ -.%S /HWXVUHZULWH<DV < δH[S>ORJ(δ9 @8 ZKHUH
9
δ
9 f
LVLQWURGXFHGWRDOORZIRUQHJDWLYHYDOXHVRI9LQ )URP WKH 500 HUURU TXDQWLOH IXQFWLRQ >(T E ZLWK η @ LQWURGXFLQJ 0 ε1= 9, 1+ ε2= 8 QDPHO\ ε1 LV VWDQGDUG QRUPDO DQG ε2 LVQRUPDOZLWKPHDQDQGXQLWYDULDQFH ZHREWDLQ
< H[S^ αλ > 9 λ±@`8
)RU λ α WKH 500 PRGHO ZLWK WKH LQWURGXFWLRQ RI δ LV HTXLYDOHQWWRWKH&DXFK\GLVWULEXWHG< *HQHUDOL]HG,QYHUVH*DXVVLDQGLVWULEXWLRQDQGWKH/HY\ GLVWULEXWLRQ /HW ' EH WKH GLVWDQFH WUDYHOHG E\ D SDUWLFOH LQ WLPH LQWHUYDO7 ZKHQ LW PRYHV DORQJ D OLQH ZLWK D XQLIRUP YHORFLW\ 9 ,I WKH SDUWLFOH LV QRZ VXEMHFWWRDOLQHDU%URZQLDQPRWLRQ QDPHO\WKHYHORFLW\9LVUDQGRP LWLVZHOONQRZQWKDWIRUDIL[HGWLPH7 WWKHGLVWDQFHRYHUZKLFKWKH
&+$37(563(&,$/&$6(62)500
SDUWLFOH WUDYHOV LV D UDQGRP YDULDEOH KDYLQJ QRUPDO GLVWULEXWLRQ ZLWK PHDQ9WDQGYDULDQFHβWZKHUHβ LVDGLIIXVLRQFRQVWDQW ILQGGHWDLOV LQ-.%S . &RQYHUVHO\LIWKHGLVWDQFHLVFRQVWDQW'WKHQWKHWLPH 7 UHTXLUHG WR FRYHU WKLV GLVWDQFH IROORZV DQ ,QYHUVH *DXVVLDQ GLVWULEXWLRQZLWKGHQVLW\IXQFWLRQ -.%S
I W πβW 'H[S^ '9W βW `W!
7KHVWDQGDUG FDQRQLFDO WZRSDUDPHWHUGHQVLW\IXQFWLRQLV RQHRI VHYHUDOSRVVLEOHIRUPVUHIHUWR-.%SRU62S
I \ π\λ H[S^(λ/µ2)(\µ \`\λ, µ!
:ULWLQJWKHH[SRQHQWDV± 6LWPD\EHVKRZQ 62SEDVHG RQ)RONVDQG&KKLNDUD WKDW6LVGLVWULEXWHGDVχZLWKRQHGHJUHH RIIUHHGRPQDPHO\ 6 (λ/µ2)(<µ < = ZKHUH = LV D VWDQGDUG QRUPDO YDULDEOH ,W LV HDV\ WR VHH WKDW = LV D PHPEHU RI WKH 500 HUURU GLVWULEXWLRQ SXW LQ WKH 500 PRGHO η λ α DVVXPHWKDWεLVVWDQGDUGQRUPDODQGWKDWµ ε2= 7KXV WKHIXQFWLRQRIWKH,QYHUVH*DXVVLDQYDULDEOH<JLYHQE\6LVDPHPEHU RIWKH500HUURUGLVWULEXWLRQ)XUWKHUPRUH6LVGLVWULEXWHGOLNH= 7KHODWWHUKDVGHQVLW\IXQFWLRQ
I \ π H[S> \ @\\
ZKLFK LV WKH /HY\ GLVWULEXWLRQ UHIHU IRU GHWDLOV WR 62 ([F S 6LQFH = LV D PHPEHU RI WKH 500 IDPLO\ RI PRGHOV LQVHUW DV DERYHZLWKWKHUHSODFHGYDOXHRIα WKH/HY\GLVWULEXWLRQLVDOVRD VSHFLDOFDVHRIWKH500HUURUGLVWULEXWLRQ 5REHUW KDV VXJJHVWHG D SDUWLFXODU IRUP RI WKH JHQHUDOL]HG LQYHUVH*DXVVLDQGLVWULEXWLRQZLWKGHQVLW\IXQFWLRQ
I [ ^. α,µ,τ ]α} H[S^> [ µ@ τ `α > 1, τ >
ZKHUH . LV D QRUPDOL]LQJ WUDQVIRUPDWLRQ H[SUHVVHG LQ WHUPV RI D FRQIOXHQWK\SHUJHRPHWULFIXQFWLRQ)RUα τ σ;RI IROORZV WKHGLVWULEXWLRQRIWKHLQYHUVHRI8QDPHO\; 8ZKHUH8LV1 σ ,WLVHDV\WRYHULI\WKDW;LVDPHPEHURIWKH500IDPLO\RIPRGHOV 7KXVIRUWKLVVSHFLDOFDVHWKH500SURYLGHVDOVRDPHPEHURIWKH JHQHUDOL]HGLQYHUVHQRUPDOGLVWULEXWLRQDVWKHODWWHUKDVEHHQGHILQHGE\ 5REHUW )LQGGHWDLOVLQ-.% S
5(63216(02'(/,1*0(7+2'2/2*<
*HQHUDOL]HGJDPPDGLVWULEXWLRQV 6XSSRVH WKDW < > ;γ)/β]F F! KDV WKH VWDQGDUG JDPPD GLVWULEXWLRQ ZLWKSDUDPHWHUαDQGGHQVLW\IXQFWLRQ
I \ \α−1H[S \ Γ(α)\≥
7KHQ ; KDV D JHQHUDOL]HG JDPPD GLVWULEXWLRQ DV GHILQHG E\ 6WDF\ 7KHGHQVLW\IXQFWLRQLV
I [ ^F>βFαΓ(α)]} [γ Fα−1H[S^> [γ)/β]F`[≥ γ.
6SHFLDO FDVHV RI WKH JHQHUDOL]HG JDPPD YDULDEOH LQFOXGH :HLEXOO α KDOIQRUPDO GLVWULEXWLRQV α F γ DQG WKH RUGLQDU\ JDPPD GLVWULEXWLRQ F ,W FDQ EH VKRZQ WKDW ZKLOH IRU WKH JDPPD YDULDEOH < WKH VNHZQHVV DQG NXUWRVLV PHDVXUHV DUH UHVSHFWLYHO\ 6N < α.X < α -.%S IRU: ORJ < WKHUHVSHFWLYH PHDVXUHVDUH -.%S 6N : ≈ α .X : ≈ α 7KXV : LV FORVHU WR QRUPDOLW\ WKDQ WKH VWDQGDUG JDPPD 5HZULWLQJ :LQWHUPVRIWKHVWDQGDUGQRUPDOYDULDEOH=ZHKDYHLQWHUPVRIWKH JHQHUDOL]HGJDPPD;
η + δ= ORJ> ;γ)/β]&
; γH[S>ORJ β F η + δ= @
RU
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α LQ WKH 500 PRGHO LV IDU IURP WKH YDOXH α= VXJJHVWHG E\ DSSUR[LPDWLRQ UHIHU WR 6KRUH IRUGHWDLOVDQGDOVRWR&KDSWHUV DQG
&+$37(563(&,$/&$6(62)500
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²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¶V $GYDQFHG 7KHRU\ RI 6WDWLVWLFV 9 'LVWULEXWLRQ7KHRU\&KDUOHV*ULIILQ &R/WG/RQGRQ >@ 7DGLNDPDOOD 3 5 DQG -RKQVRQ 1 / 6\VWHPV RI IUHTXHQF\ FXUYHV JHQHUDWHGE\WUDQVIRUPDWLRQVRIORJLVWLFYDULDEOHV%LRPHWULND
5(63216(02'(/,1*0(7+2'2/2*<
> @7XNH\-: 0RGHUQWHFKQLTXHVLQGDWDDQDO\VLV16)VSRQVRUHGUHJLRQDO UHVHDUFK FRQIHUHQFH DW 6RXWKHDVWHUQ 0DVVDFKXVHWWV 8QLYHUVLW\ 1RUWK 'DUPRXWK 0$ >@:LOVRQ(%+LOIHUW\00 7KHGLVWULEXWLRQRIFKLVTXDUH3URFHHGLQJVRI WKH1DWLRQDO$FDGHP\RI6FLHQFHV
&KDSWHU
(YDOXDWLQJ500IRU&RPSOLDQFH
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
5(63216(02'(/,1*0(7+2'2/2*<
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
0HWKRGRORJ\ /LQHDUUHJUHVVLRQ %&7UDQVIRUPDWLRQ */0
500
&+$37(5(9$/8$7,1*500)25&203/,$1&(
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λ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
5(63216(02'(/,1*0(7+2'2/2*<
WUDQVIRUPDWLRQZKHUHWKHSDUDPHWHUλHVWLPDWHGIURPGDWDDQDO\VLVLVD VKDSHSDUDPHWHUWKDWGHWHUPLQHVWKHVWUXFWXUHRIWKHILQDOPRGHO ,Q 500 WZR SDUDPHWHUV GRPLQDWH LQ GHWHUPLQLQJ WKH PRGHO V VWUXFWXUH α DQG λ IRU H[WUHPHO\ FRQYH[ UHODWLRQVKLSV PRUH WKDQ WZR SDUDPHWHUV PD\ EH QHHGHG DV H[SODLQHG LQ &KDSWHU $YDLODELOLW\ RI WZR SDUDPHWHUV WR GHWHUPLQH WKH VKDSH RI WKH PRGHO LQ DGGLWLRQ WR D VFDOH SDUDPHWHU µ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λDSSURDFKHVXQLW\7KXV DSSUR[LPDWH QRUPDOLW\ IRU WKH HUURU GLVWULEXWLRQ DV D UHVXOW RI ODUJH LQFUHDVH LQ WKH OLQHDU SUHGLFWRU LV QRW UHQGHUHG E\ WKLV PRGHOLQJ DSSURDFK 1HLWKHULV GHFRXSOLQJ RIWKH YDULDQFH IURP WKH PHDQ LPSOLHG E\LW ,PSOHPHQWDWLRQ RI */0 UHTXLUHV WKDW WKH HUURU GLVWULEXWLRQ LV VSHFLILHG LQ DGYDQFH DQG FRQVHTXHQWO\ VR LV WKH UHODWLRQVKLS EHWZHHQ WKH YDULDQFH DQG WKH PHDQ 6LQFH WKH ODWWHU UHODWLRQVKLS QHHGHG WR GHWHUPLQHZHLJKWVLQWKHLWHUDWLYHO\UHZHLJKWHGOHDVWVTXDUHVHVWLPDWLRQ
&+$37(5(9$/8$7,1*500)25&203/,$1&(
URXWLQH LV VSHFLILHG LQ DGYDQFH QR GHFRXSOLQJ RI WKHVH GLVWULEXWLRQDO SDUDPHWHUVLVLQGHHG SRVVLEOH */0 LVIDU IURP VDWLVI\LQJ 5HTXLUHPHQW 500GLIIHUVLQDPDMRUZD\6LQFHLQ500ERWKαDQGλGHWHUPLQH WKH VWUXFWXUH RI WKH PRGHO DQG VLQFH WKH OLQHDU SUHGLFWRU LV DVVRFLDWHG ZLWKDVHSDUDWHHUURUε1WKHUHODWLYHVL]HRIWKLVHUURU UHODWLYHWRη)ZLOO GHWHUPLQH ZKHWKHU FKDQJHV LQ WKH V\VWHPDWLF YDULDWLRQ FRPSRQHQW FKDQJHV LQ η ZRXOG DIIHFW WKH VKDSH RI WKH HUURU GLVWULEXWLRQ $V η EHFRPHV ODUJHU ε WHQGV WR EH QHJOLJLEOH DQG WKH YDULDQFH RI WKH HUURU GLVWULEXWLRQ ZLOO WHQG WR VWDELOL]H 7KH HUURU GLVWULEXWLRQ ZLOO WKHQ EH GHWHUPLQHG PDLQO\ E\ WKH RWKHU HUURU WHUP ε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αλ > ηε1 λ±@µε2`
H[S^ αλ > ησε1= λ±@µσε2[ρ= ρ =@`
: αλ > ησε1= λ±@µσε2[ρ= ρ =@
ZKHUHε1DQGε2DUHHUURUVGHULYHGIURPDELYDULDWHQRUPDOGLVWULEXWLRQ ZLWKFRUUHODWLRQρ%\FRQGLWLRQLQJ=_= ]ZHKDYHREWDLQHGIRUWKH HUURUV ε1 σε1= ε2 σε2[ρ= ρ =@ ZKHUH=DQG=DUHLQGHSHQGHQWVWDQGDUGQRUPDOYDULDEOHV
5(63216(02'(/,1*0(7+2'2/2*<
$VVXPLQJWKDWWKHOLQHDUSUHGLFWRULVFRQVWDQWDUELWUDULO\VHWWLQJLWWR RQHRQHREWDLQVIURP DQG < H[S^ αλ > 1ε1 λ±@µε2`
H[S^ αλ > 1σε1= λ±@µσε2[ρ= ρ =@`
: αλ > σε1= λ±@µσε2[ρ= ρ =@
9DULDWLRQVRIWKHEDVLFPRGHODVVXPLQJGLIIHUHQWHUURUVWUXFWXUHVDUH UHIHUWR&KDSWHU IRUGHWDLOV < H[S^ αλ > ε1 λ±@µ2` ε2) H[S^ αλ > 1σε1= λ±@µ`^σε2[ρ= ρ =@` < H[S^ αλ >H[S λε1)±@µε2} H[S^ αλ >H[S λσε1=)±@µσε2[ρ= ρ =@` < H[S^ αλ >H[S λε1)±@µ` ε2) H[S^ αλ >H[S λσε1=)±@µ`^σε2[ρ= ρ =@` %\FRQGLWLRQLQJ=_= ]IRXUHTXLYDOHQWIRUPVRI FDQ VLPLODUO\EHGHYHORSHG ILQGGHWDLOVLQ&KDSWHU /HWXVH[DPLQHKRZZHOOWKH500HUURUGLVWULEXWLRQZLWKYDULDWLRQV JLYHQ E\ IXOILOOV WKH UHTXLUHPHQWV WKDW UHODWH WR PRGHOLQJ UDQGRPYDULDWLRQ 5HTXLUHPHQWVDQG 5HTXLUHPHQWDVVHUWVWKDWDWOHDVWWKHILUVWWKUHHRUIRXUPRPHQWVRI WKH XQGHUO\LQJ HUURU GLVWULEXWLRQ EH SUHVHUYHG LQ WKH HPSLULFDOO\ ILWWHG PRGHO$VH[SODLQHGHDUOLHU &KDSWHU WKLVUHTXLUHVWZRIHDWXUHV)LUVW WKDW WKHUH DUH HQRXJK SDUDPHWHUV DSDUW IURP WKRVH DVVRFLDWHG ZLWK WKH OLQHDU SUHGLFWRU DVVXPHG QRZ WR EH FRQVWDQW 7KLV ZRXOG DOORZ IOH[LELOLW\LQUHIOHFWLQJYDULRXVO\VKDSHGGLVWULEXWLRQV6HFRQGO\WKDWWKH DOOLHG HVWLPDWLRQ SURFHGXUH JXDUDQWHHV WKDW WKH HVWLPDWHG TXDQWLOH IXQFWLRQ SUHVHUYHV ZHOO ORZ 06(V DW OHDVW WKH ILUVW WKUHH RI IRXU PRPHQWVRIWKHWUXHHUURUGLVWULEXWLRQ 7KH QHZ 500 PRGHO DWWDLQV ERWK IHDWXUHV :LWK UHVSHFW WR WKH ILUVW IHDWXUH WKH WKUHH SDUDPHWHUV ^α λ µ` SURYLGH WKH IOH[LELOLW\ QHHGHG WR DFFRPPRGDWH YDULRXVO\VKDSHG HUURU GLVWULEXWLRQV 7KLV KDV EHHQ GHPRQVWUDWHG IRU D ODUJH VHW RI GLVWULEXWLRQV ZLWK GLIIHUHQW VKDSH FKDUDFWHULVWLFV LQ 6KRUH IRU WKH LQYHUVH QRUPDOL]LQJ
&+$37(5(9$/8$7,1*500)25&203/,$1&(
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√β1, β2 SODQH7KHLVVXHRIFRYHUDJHZDVDGGUHVVHGLQ6KRUH DQG LQ &KDSWHU :KLOH XQDEOH IRU FRPSXWDWLRQDO UHDVRQV WR GHULYH D FRPSUHKHQVLYH PDS RI DUHDV LQ WKH √β1, β2 SODQH RFFXSLHG E\ WKH 500 HUURU GLVWULEXWLRQ ZH KDYH GHPRQVWUDWHG XVLQJ GLIIHUHQW YDOXHV IRUWKHSDUDPHWHUVRIWKLVGLVWULEXWLRQWKDWDODUJHVSHFWUXPRI √β1, β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√β1, β2 SODQHRFFXSLHGE\WKH-RKQVRQIDPLO\RIGLVWULEXWLRQV &RQWUDVWHG ZLWK KRZ FXUUHQW DSSURDFKHV WR PRGHO UDQGRP YDULDWLRQ FRPSO\ ZLWK 5HTXLUHPHQWV DQG UHIHU WR DQ HDUOLHU GLVFXVVLRQ LQ &KDSWHUVDQG LWVHHPVWKDWDGRSWLQJWKH500HUURUGLVWULEXWLRQDVD SODWIRUPIRUPRGHOLQJUDQGRPYDULDWLRQFDQSURYLGHEHQHILWVQRWZLGHO\ HQWHUWDLQHGE\FXUUHQWPHWKRGRORJLHV
5(63216(02'(/,1*0(7+2'2/2*<
5HIHUHQFHV >@ 6KRUH + $ QHZ DSSURDFK WR DQDO\VLQJ QRQQRUPDO TXDOLW\ GDWD ZLWK DSSOLFDWLRQ WR SURFHVV FDSDELOLW\ DQDO\VLV ,QWHUQDWLRQDO -RXUQDO RI 3URGXFWLRQ 5HVHDUFK ,-35 >@ 6KRUH + *HQHUDO FRQWURO FKDUWV IRU YDULDEOHV ,QWHUQDWLRQDO -RXUQDO RI 3URGXFWLRQ5HVHDUFK >@6KRUH+ 5HVSRQVH0RGHOLQJ0HWKRGRORJ\ 500 ([SORULQJWKHLPSOLHG HUURU GLVWULEXWLRQ &RPPXQLFDWLRQV LQ 6WDWLVWLFV 7KHRU\ DQG 0HWKRGV >@ 6KRUH + D 5HVSRQVH 0RGHOLQJ 0HWKRGRORJ\ 500 9DOLGDWLQJ HYLGHQFH IURP HQJLQHHULQJ DQG WKH VFLHQFHV 4XDOLW\ DQG 5HOLDELOLW\ (QJLQHHULQJ ,QWHUQDWLRQDO >@6KRUH+ E 5HVSRQVH0RGHOLQJ0HWKRGRORJ\ 500 &XUUHQWGLVWULEXWLRQV WUDQVIRUPDWLRQV DQG DSSUR[LPDWLRQV DV VSHFLDO FDVHV RI WKH 500 HUURU GLVWULEXWLRQ &RPPXQLFDWLRQV LQ 6WDWLVWLFV 7KHRU\ DQG 0HWKRGV >@ 6KRUH + F 1RQQRUPDO SRSXODWLRQV LQ TXDOLW\ DSSOLFDWLRQV $ UHYLVLWHG SHUVSHFWLYH4XDOLW\DQG5HOLDELOLW\(QJLQHHULQJ,QWHUQDWLRQDO
3DUW,,,
02'(/,1*6<67(0$7,& 9$5,$7,21$33/,&$7,216
This page intentionally left blank
&KDSWHU
&RPSDUDWLYH6ROXWLRQVIRU5HODWLRQDO0RGHOV
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
5(63216(02'(/,1*0(7+2'2/2*<
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
5XQ
<
$
%
&
'
JRRGSDUWV
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
QDWXUHRIWKHUHVSRQVH7KH\FRQFOXGHGWKDWWKHPDLQHIIHFWV$&DQG' DUHVLJQLILFDQW 7KHGDWDZHUHUHDQDO\]HGE\+DPDGDDQG1HOGHU +1 LQWKH IUDPHZRUNRI*/0XVLQJDORJLWOLQNIXQFWLRQDQGDVVXPLQJDELQRPLDO GLVWULEXWLRQ7KH\GHULYHGWKHIROORZLQJPRGHO ORJLW 3 ORJ>3 3 @
$ &
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
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH([DPSOH5HVXOWVIURP&&$ZLWKGLIIHUHQWQXPEHURIWHUPVLQWKH7D\ORU DSSUR[LPDWLRQWR* < RSWLPDOFKRLFHLVVWDUUHG <7LPHDQVWHUPVXSWRDQG LQFOXGLQJ<7L &DQ&RUUHODWLRQV 6LJQLILFDQFH )LUVW&DQ&RU 6LJQLILFDQFH 6HFRQG&DQ&RU 6LJQLILFDQFH
7D\ORUH[SDQVLRQXSWRWHUP <7 <7
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
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
LQRXUPRGHO,QIDFWDOOHIIHFWVLQ KDYHVLJQLILFDQFHOHYHORIWKH VDPHRUGHURIPDJQLWXGHDVWKDWIRU$& QDPHO\S *LYHQ WKH /3 LQ 0/ HVWLPDWHV IRU WKH 500SDUDPHWHUV FDQ QRZ EH REWDLQHG 7KH HVWLPDWHG 500 PRGHO ZRXOG SURYLGH WKH QRQ OLQHDUUHODWLRQVKLSEHWZHHQWKH/3DQGWKHUHVSRQVH 7KH ODWWHU PD\ EH WKH SURSRUWLRQ 3 DV XVHGLQ GHULYLQJ WKH /3 RU WKHDFWXDOIUHTXHQF\3 DVJLYHQLQ7DEOH :HSURFHHGZLWK3 )URP DQG ZHKDYH <7<7
$& $&ε
5HFDOOIURP&KDSWHUWKDWLQHVWLPDWLQJDQ500PRGHOWZRVHWVRI SDUDPHWHUV QHHG WR EH HVWLPDWHG E\ WZR GLIIHUHQW SURFHGXUHV :H XVH ZHLJKWHG QRQOLQHDU OHDVWVTXDUHV :1//6 WR HVWLPDWH WKH QRQ OLQHDU SDUDPHWHUV ^α,λ,µ2` 6HW ,, DQG ZH XVH D ORJOLNHOLKRRG IXQFWLRQEDVHGRQWKHGHQVLW\IXQFWLRQ GI RIWKHWUDQVIRUPHGUHVSRQVH : ORJ < WRHVWLPDWHWKHHUURUSDUDPHWHUV^ρ, σε1, σε2}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η > 3µ3 σ3@> 3µ3 σ3@
$& $&
ZLWK µ3 DQG σ3 EHLQJ WKH PHDQ DQG WKH VWDQGDUG GHYLDWLRQRI39DOXHVRIWKHSUHGLFWHG3IRULQGLYLGXDOREVHUYDWLRQVPD\ EHH[WUDFWHGIURP HLWKHUYLDDURRWILQGLQJURXWLQHRUE\VROYLQJ WKH TXDGUDWLF HTXDWLRQ H[SOLFLWO\ IRU 3 3XUVXLQJ WKH ODWWHU RSWLRQ WZR VROXWLRQV ZLOO EH REWDLQHG IRU HDFK REVHUYDWLRQ 2QO\ RQH RI WKHVH SURYLGHVDVDWLVIDFWRU\SUHGLFWHGYDOXH
5(63216(02'(/,1*0(7+2'2/2*<
)RUWKH+1PRGHO3LVH[WUDFWHGIURP>UHODWHWR @ 3 >H[S η @
ZKHUHηLVJLYHQE\WKH5+6RI $V D WKLUG PRGHOLQJ HIIRUW ZH RSWHG WR XVH ORJLW DV D WUDQVIRUPHG UHVSRQVHLQVWHSZLVHOLQHDUUHJUHVVLRQWRREWDLQWKHIROORZLQJPRGHO ORJ>3 3 @ ± $±& 7KLV PRGHO KDV 5DGM ZLWK )UDWLR RI :LWK DQG GHJUHHVRIIUHHGRPIRUWKHPRGHODQGWKHHUURUUHVSHFWLYHO\WKLVUHVXOWLV KLJKO\VLJQLILFDQW S &RPSDULQJ WKH UHVLGXDO HUURU VWDQGDUG GHYLDWLRQV IRU DOO WKUHH PRGHOVZHREWDLQ )RUWKH*/0PRGHO> DQG @ σUHV )RUWKH500PRGHO σUHV )RUWKHGDWDWUDQVIRUPHGORJLWPRGHO σ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λRIWKH %R[&R[ WUDQVIRUPDWLRQ WKLV LV FRPPRQO\ UHJDUGHG DV RIIOLQH QRW FRQVLGHUHGLQWKHGHWHUPLQDWLRQRIWKHPRGHO VGHJUHHVRIIUHHGRPZKHQ OLQHDUUHJUHVVLRQLVVXEVHTXHQWO\DSSOLHG )RU D VPDOO VDPSOH VL]H WKLV FRPPRQO\ DFFHSWHG LJQRULQJ RI RII OLQHGHFLVLRQVDQGHVWLPDWLRQPD\QHHGWREHJLYHQDKLJKHUZHLJKW IRU ERWK500*/0DQGWKH%&WUDQVIRUPDWLRQ WKDQFXUUHQWO\SUDFWLFHG
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
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
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH([DPSOH9DULDEOHVIRUWKH%LJ0DF3DULW\,QGH[ 9DULDEOH1DPH
'HVFULSWLRQ
%LJ0DF %0 5HVSRQVH
0LQXWHVRIODERUUHTXLUHGE\DQDYHUDJH ZRUNHUWREX\D%LJ0DFDQG)UHQFKIULHV
%UHDG %
0LQXWHVRIODERUUHTXLUHGWREX\RQHNLORJUDP RIEUHDG
%XV)DUH %)
7KHORZHVWFRVWRIDWHQNLORPHWHUEXVWUDP RUVXEZD\WLFNHWLQ86GROODUV
(QJ6DO (6
7KHDYHUDJHDQQXDOVDODU\RIDQHOHFWULF HQJLQHHULQWKRXVDQGVRI86GROODUV
(QJ7D[ (7
7KHDYHUDJHWD[UDWHSDLGE\HQJLQHHUV
6HUYLFH 6
$QQXDOFRVWRI VHUYLFHVSULPDULO\UHOHYDQW WR(XURSHDQG1RUWK$PHULFD
7HDFK6DO 76
7KHDYHUDJHDQQXDOVDODU\RIDSULPDU\VFKRRO WHDFKHULQWKRXVDQGVRI86GROODUV
7HDFK7D[ 77
7KHDYHUDJHWD[UDWHSDLGE\SULPDU\WHDFKHUV
9DF'D\V 9'
$YHUDJHGD\VRIYDFDWLRQSHU\HDU
:RUN+UV :+
$YHUDJHKRXUVZRUNHGSHU\HDU
&LW\
1DPHRIFLW\
7DEOH([DPSOH5HVXOWVIURP&&$ZLWKDGLIIHUHQWQXPEHURIWHUPVLQWKH7D\ORU DSSUR[LPDWLRQWR* < WKHRSWLPDOFKRLFHLVVWDUUHG <7LPHDQVWHUPVXSWRDQG LQFOXGLQJ<7L 6WDWLVWLF
<7
<7
<7
<7
5 &DQRQ&RUUVTXDUHG
6LJQLILFDQFH
6LJ UHPRYHG
6LJ UGUHPRYHG
VW
6LJ URRWUHPRYHG QG
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
,WLVREYLRXVIURP7DEOHWKDWVHOHFWLQJWHUPVXSWRDQGLQFOXGLQJ <7 QDPHO\ D TXDGUDWLF HTXDWLRQ GHOLYHUV WKH EHVW 7D\ORU DSSUR[L PDWLRQIRUWKHWUDQVIRUPHGUHVSRQVH 7KHDVVRFLDWHGH[SUHVVLRQIRUWKHFDQRQLFDOVFRUHV>WKHDSSUR[LPDWLRQ IRU* < @LV : <7<7 7KH FDQRQLFDO VFRUHV FDOFXODWHG IURP IRU LQGLYLGXDO REVHUYDWLRQV DUH QRZ LQFRUSRUDWHG LQ D VWHSZLVH PXOWLSOH OLQHDU UHJUHVVLRQ DQDO\VLV 7KH LQLWLDO PRGHO IRU HIIHFWV WR EH LQFOXGHG LQ WKH DQDO\VLVLVIDFWRULDORIVHFRQGGHJUHH 'LDJQRVWLFFKHFNLQJRIWKHUHVXOWLQJPRGHOVKRZVWKDWRXWRIWKH FLWLHV WKDW SDUWLFLSDWH LQ WKH DQDO\VLV VHYHQ FLWLHV KDYH VWXGHQWL]HG UHVLGXDOV WKDW H[FHHG DERXW ± 7KHVH DUH RUGHU QXPEHU QDPH VWXGHQWL]HGUHVLGXDO ^ $WKHQV %XHQRV$LUHV /X[HPERXUJ 0DGULG 0H[LFR&LW\ 6DR3DXOR 7RN\R ` $QHJDWLYHUHVLGXDOLPSOLHVWKDWWKHUHVSRQVHLVVPDOOHUWKDQWKHPRGHO VXJJHVWV ([FOXGLQJ WKH DERYH IURP SXUVXLQJ DQDO\VHV ZH REWDLQ WKH ILQDO PRGHOIRU/3 VLJQLILFDQFHOHYHODSSHDUVEHORZHDFKHIIHFWLQEUDFNHWV
/3 % %) % %)
S S S
6 ± 6 %)
S S
76
S
1RWHWKDWWKHLQWHUFHSWLV]HURHGVLQFHLWZDVQRWVLJQLILFDQW 7R DVVHVV LQ DQ DGHTXDWH PDQQHU WKH LQIOXHQFH RI WKH YDULRXV SUHGLFWRUV ZKLFKDUHPHDVXUHGRQGLIIHUHQWVFDOHV WKH/3LVSUHVHQWHG
5(63216(02'(/,1*0(7+2'2/2*<
ZLWKVFDOHGHVWLPDWHV VFDOLQJLVSHUIRUPHGE\VXEWUDFWLQJWKHPHDQDQG GLYLGLQJE\KDOIWKHUDQJHLQWKHGDWD WRJLYH /3 % %) % %)
6 6 %) 76
2QHFDQOHDUQIURP WKDWWKHLQWHUDFWLRQVSOD\LPSRUWDQWUROHLQ PRGHOLQJ%LJ0DF$OVRJLYHQWKHPDLQHIIHFWVDQGWKHLQWHUDFWLRQVWKDW WXUQHGRXWDVVLJQLILFDQWWKLVPRGHOPDNHVVHQVHRQLQWXLWLYHJURXQGV 5HJDUGLQJ JRRGQHVVRIILW VWDWLVWLFV WKH PRGHO KDV 5DGM RI ZLWK UHVLGXDO VWDQGDUG GHYLDWLRQ RI 7KH PRGHO )UDWLR LV ZKLFK ZLWK PRGHO GI RI νң= DQG HUURU GI RI ν LV KLJKO\ VLJQLILFDQW SRIWKHRUGHURI :HSURFHHGWR3KDVHRIWKHDQDO\VLVDQGHVWLPDWHWKHSDUDPHWHUVRI WKH500PRGHO7KHODWWHUZLOOUHODWHWKH/3WRWKHUHVSRQVH%LJ0DF 7DEOH 7RDYRLGQHJDWLYHYDOXHVLQWKH/3WKHPLQLPDOYDOXH LVILUVW DGGHG WR HDFK REVHUYDWLRQ DMXVWLILFDWLRQ IRUWKLV ZDV JLYHQ LQ &KDSWHU $GRSWLQJWKHEDVLF500PRGHODQGDVVXPLQJIRUVLPSOLFLW\WKDWWKH HUURUV DUH LQGHSHQGHQW ρ=), DQ XQZHLJKWHG QRQOLQHDU OHDVWVTXDUHV 1//6 LV ILUVW DSSOLHG ZLWK WKH PHDQ ( : JLYHQ E\ D $Q HVWLPDWH IRU λ LV REWDLQHG YHU\ QHDU λ= DQG DQ HVWLPDWH IRU σε1 YHU\QHDU]HUR,WLVWKHUHIRUHDVVXPHGWKDWλ=1 DQG σε1 1RWHWKDW WKH VPDOO YDOXH RI σε1 LPSOLHV WKDW D ORJWUDQVIRUPHG UHVSRQVH KDV D FRQVWDQWYDULDQFHDQGWKHUHIRUHQRZHLJKWLQJLQWKH1//6LVZDUUDQWHG $GGLWLRQDOO\ WKH YDOXH RI λ LPSOLHV WKDW D ORJ WUDQVIRUPDWLRQ RI WKH UHVSRQVHZRXOGGHOLYHUDGGLWLYHHIIHFWVDQGWKHUHIRUHWKHUHLVQRQHHGWR SURFHHG WR WKH VHFRQG VWDJH RI WKH DQDO\VLV ZKHUH WKH 500 HUURU SDUDPHWHUVDUHHVWLPDWHG $UHUXQRIXQZHLJKWHG1//6ZLWKWKHDERYHYDOXHVJLYHV α µ2 σ2UHV
6RPHJRRGQHVVRIILWVWDWLVWLFVUHODWLQJWRWKHVHHVWLPDWHVDUHJLYHQLQ 7DEOH *LYHQ WKH DVVXPSWLRQ WKDW σε1 RQH FDQ FRQVLGHU WKH UHVLGXDOYDULDQFHσ2UHVREWDLQHGLQWKH1//6SURFHGXUHDVDQHVWLPDWH RIWKHSDUDPHWHUσε22.
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
)LJXUH GLVSOD\V D VFDWWHU SORW RI WKH UHODWLYH HUURUV ZKHUH WKH ODWWHUDUHFDOFXODWHGRQWKHRULJLQDOVFDOHRIWKHUHVSRQVH %LJ0DF :H UHDOL]HWKDWIRUPRVWRIWKHGDWDWKHUHODWLYHHUURUGRHVQRWH[FHHGDERXW ± 7RDSSUHFLDWHWKHTXDOLW\RIWKHPRGHOLQJHIIRUWGHYHORSHGDERYHZH UHODWH WR WKH VROXWLRQ JLYHQ LQ &: 7KH\ DGYRFDWH WKHUHLQ S WUDQVIRUPLQJERWKWKHUHVSRQVHDQGWKHSUHGLFWRUV (6 DQG 76 6SHFLILFDOO\WKHVXJJHVWHGPRGHOLV %0 β β (6 β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ν1= DQG HUURU GI RI ν 7KLV)YDOXHLVKLJKO\VLJQLILFDQW S )LJXUH GLVSOD\V D VFDWWHU SORW RI WKH UHODWLYH HUURUV ZKHUH WKH ODWWHUDUHFDOFXODWHGLQWKHRULJLQDOVFDOHRIWKHUHVSRQVH %LJ0DF 7DEOH([DPSOH)LQDOHVWLPDWHVRIWKH6WUXFWXUDOSDUDPHWHUV^αµ`ZLWK VWDQGDUGHUURUVDQG FRQILGHQFHLQWHUYDOV
3DUDPHWHU
(VWLPDWH
6(
&,
α µ
^ `
^`
5(63216(02'(/,1*0(7+2'2/2*<
)LJXUH ([DPSOH $ VFDWWHU SORW RI WKH UHODWLYH HUURUV Q IRU WKH %LJ 0DF H[DPSOH 500$QDO\VLV +RUL]RQWDOD[LVLV%LJ0DFLQWKHRULJLQDOVFDOH
)LJXUH ([DPSOH $ VFDWWHU SORW RI WKH UHODWLYH HUURUV Q IRU WKH %LJ 0DF H[DPSOH &:PRGHO +RUL]RQWDOD[LVLV%LJ0DFLQWKHRULJLQDOVFDOH
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
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
ORJ µ $%& '(*
$& %'
0\HUVHWDO GHULYHDGLIIHUHQWPRGHOXVLQJWKHVDPHYDULDQFH IXQFWLRQ DQG GLIIHUHQW OLQN IXQFWLRQV KRZHYHU DFFRXQWLQJ IRU RYHU
5(63216(02'(/,1*0(7+2'2/2*<
GLVSHUVLRQ)RUWKHORJOLQNWKH\REWDLQ WKHUHLQS
ORJ µ &*$&%' D )RUWKHVTXDUHURRWOLQNWKH\REWDLQ WKHUHLQS µ $% & (*
$& %'
E
:H UHDQDO\]H WKHVH GDWD VWDUWLQJ ZLWK VLPSOH OLQHDU UHJUHVVLRQ SURFHHGLQJZLWKD%&WUDQVIRUPDWLRQDQGFRQFOXGLQJE\UHLQWURGXFLQJ UHVXOWVIURPWKH500VROXWLRQ 6HFWLRQ $SSO\LQJVWHSZLVHOLQHDUUHJUHVVLRQZLWKDQLQLWLDOIDFWRULDOPRGHORI GHJUHHWZRWKHIROORZLQJPRGHOLVREWDLQHG /3 $% & (*
$ & % '
ZLWK5 DGM DQGPRGHO)UDWLRRI/DFNRIILWWHVWSURYHV WREHKLJKO\VLJQLILFDQW S ZKLFKLQYDOLGDWHWKLVPRGHO $SSO\LQJD%&WUDQVIRUPDWLRQWRWKHUHVSRQVHUHVXOWVLQ < λ) <
)
QRWHWKDWWKLVLVDQRUPDOL]HG%&WUDQVIRUPDWLRQZKHUHWKHGHQRPLQDWRU FRQWDLQVWKHJHRPHWULFDYHUDJHRIWKHWUDQVIRUPHGYDOXHV 1RUPDOL]DWLRQ LV VWDQGDUG SUDFWLFH WKDW DOORZV FRPSDULVRQ RI GLIIHUHQWO\VFDOHG %& WUDQVIRUPDWLRQVUHIHUWR&KDSWHUIRUGHWDLOV )URPOLQHDUUHJUHVVLRQDSSOLHGWRWKHWUDQVIRUPHGUHVSRQVHZHREWDLQ (><(λ)@ /3 &*
$ & % '
1RWH WKDW VLQFH WKH IDFWRUV DUH JLYHQ LQ VWDQGDUGL]HG RU QRPLQDO YDOXHVRIDQGWKHUHLVQRPHDQLQJWRFHQWHULQJE\VXEWUDFWLQJ WKH PHDQ 7KLV RSHUDWLRQ LV DXWRPDWLFDOO\ FRQGXFWHG E\ WKH FRPSXWHULQGLVSOD\LQJLQWHUDFWLRQVDQGZHSUHVHQWWKHUHVXOWVH[DFWO\DV REWDLQHGIURPWKHFRPSXWHURXWSXW$OVRQRWHWKDWHVWLPDWHGHIIHFWVZLWK VLJQLILFDQFH OHYHO RI DW PRVW ZHUH LQFOXGHG LQ ERWK RI WKH DERYH PRGHOV :LWK5DGM DQGPRGHO)UDWLRRI LQWURGXFHV YHU\ OLWWOH LPSURYHPHQW LQ JRRGQHVVRIILW VWDWLVWLFV UHODWLYH WR
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
+RZHYHU WKH WUDQVIRUPHG UHVSRQVH UHVXOWV LQ D VLPSOHU PRGHO ILYH SDUDPHWHUV DUH HVWLPDWHG LQVWHDG RI HLJKW $ WHVW RI ODFNRIILW VWLOO SURYHVWREHPLOGO\VLJQLILFDQW S 3URFHHGLQJZLWKWKH500DSSURDFKWKH/3GHULYHGLQ6HFWLRQLV /3 $%&( *
$ & % '
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
5(63216(02'(/,1*0(7+2'2/2*<
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µ ;; ;; ;;;;;;
,W LV LQWHUHVWLQJ WR QRWLFH WKDW ZKLOH IRU ; WKH VLJQLILFDQFH OHYHO LV S IRUDOORWKHUPDLQHIIHFWV ;;DQG; WKHVLJQLILFDQFHOHYHO LVS 5HDQDO\]LQJ WKH GDWD D %& WUDQVIRUPDWLRQ LV ILUVW DSSOLHG DQGWKHQ WKH500VROXWLRQGHULYHGLQ6HFWLRQLVDGGUHVVHG $%&WUDQVIRUPDWLRQEDVHGRQDPRGHOZLWKWKHHIIHFWV^;;; ;;;;`ZKHUHVLJQLILFDQFHOHYHORIS LVUHTXLUHGIRULQFOXVLRQ LQWKHPRGHOLV <(λ) <
8VLQJ WKH WUDQVIRUPHG < DV D UHVSRQVH ZH REWDLQ IURP VWHSZLVH OLQHDUUHJUHVVLRQ (><(λ)] ;;;
;;;;
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
7KH JRRGQHVVRIILW VWDWLVWLF LV 5DGM ZLWK PRGHO )UDWLR RI +RZHYHURQO\;DQG;DUHKLJKO\VLJQLILFDQW S 3URFHHGLQJZLWK500ZHREWDLQHGLQ6HFWLRQ /3 ;; ; ;;;;;;
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
5(63216(02'(/,1*0(7+2'2/2*<
&RPSDULVRQRI0RGHOV 0DOORZ V&S 'LIIHUHQWVWDWLVWLFVPD\EHXVHGIRUWKHVHOHFWLRQRIWKHEHVWPRGHO,Q WKLV VXEVHFWLRQ ZH LQWURGXFH EULHIO\ 0DOORZ V &S FULWHULRQ $NDLNH V ,QIRUPDWLRQ&ULWHULRQ $,& ZLOOEHWUHDWHGLQ6XEVHFWLRQ 0DOORZ KDVVXJJHVWHGXVHRIWKH&SVWDWLVWLFIRUOLQHDU UHJUHVVLRQ &S 566S6 QS ZKHUH 566 LV WKH UHVLGXDO VXP RI VTXDUHV IURP D PRGHO FRQWDLQLQJ S SDUDPHWHUV >LQFOXGLQJ β WKH LQWHUFHSW WKH FRQVWDQW @ DQG 6 LV WKH UHVLGXDO PHDQ VTXDUH IURP WKH FRPSOHWH PRGHO WKH PRGHO WKDW LQFOXGHV DOO SRVVLEOH SUHGLFWRU YDULDEOHV ,W LV DVVXPHG WKDW WKH PRGHO ZLWKSSDUDPHWHUVLVDVXEPRGHORIWKHFRPSOHWHPRGHO6LVDVVXPHG WREHDUHOLDEOHXQELDVHGHVWLPDWHRIWKHWUXHHUURUYDULDQFHσε2QDPHO\ ( 6 σε2$VWHUPVDUHDGGHGWRDPRGHO566ZLOOGHFUHDVHEXW&SZLOO XVXDOO\LQFUHDVH ,ID PRGHOLVDGHTXDWHWKDWLVGRHVQRWVXIIHUIURPODFNRIILWWKHQ ( 566S QS σε2, DQGZHKDYHIRUDQDGHTXDWHPRGHO ( &S ( 566S ( 6 QS QS σε2 σε2 QS S
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
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
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α, λ, µ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
5(63216(02'(/,1*0(7+2'2/2*<
7RFRPSDUHWZRPRGHOVVD\DVLPSOHPRGHO$ZLWKS$SDUDPHWHUV DQGDPRUHFRPSOLFDWHGPRGHO%ZLWKDODUJHUQXPEHURISDUDPHWHUV S%!S$WKHGLIIHUHQFHLQWKH$,&YDOXHVLVWKHPHDQLQJIXOVWDWLVWLF ∆$,& $,&%$,&$ QORJ 566%566$ S%S$
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
&+$37(5&203$5$7,9(62/87,216)255(/$7,21$/02'(/6
>@0\HUV5+0RQWJRPHU\'& $WXWRULDORQJHQHUDOL]HGOLQHDUPRGHOV -RXUQDORI4XDOLW\7HFKQRORJ\ >@0\HUV5+0RQWJRPHU\'&9LQLQJ** *HQHUDOL]HG/LQHDU0RGHOV ZLWK$SSOLFDWLRQVLQ(QJLQHHULQJDQGWKH6FLHQFHV-RKQ:LOH\ 6RQV
This page intentionally left blank
&KDSWHU
5HOLDELOLW\(QJLQHHULQJ ZLWK&HQVRULQJ
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
5(63216(02'(/,1*0(7+2'2/2*<
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η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ηLQ3KDVHRIWKHHVWLPDWLRQSURFHGXUH RQO\FRPSOHWH XQFHQVRUHG REVHUYDWLRQVZHUHXVHG7KLVLVDUHVWULFWLYH DVVXPSWLRQ ZKLFK LV QRW QHFHVVDULO\ WUXH +RZHYHU LW VLPSOLILHV DSSUHFLDEO\WKHGHYHORSHGSURFHGXUH 7KHEDVLF500PRGHOLV < H[S^ αλ > ηε1 λ@µε2` H[S^ αλ > ησε1Ζ1 λ@µσε2[ρ= ρ =@`
λ
: ORJ < αλ > ησε1Ζ1 @µ σε2[ρ= ρ =@ ZKHUHε1DQGε2 DUHFRUUHODWHGHUURUVDQG=DQG=DUHWZRLQGHSHQGHQW XQFRUUHODWHG VWDQGDUGQRUPDOYDULDEOHV
&+$37(55(/,$%,/,7<(1*,1((5,1* :,7+&(1625,1*
)URP WKH H[SUHVVLRQ IRU WKH GHQVLW\ IXQFWLRQ GI RI : I: Z θ θ >(TXDWLRQ @ WKH OLNHOLKRRG IXQFWLRQ / IRU D JLYHQ VDPSOH RI Q FRPSOHWH XQFHQVRUHG REVHUYDWLRQV^ ηZ ηZ « ηQZQ `LV U
θ Iε1 ε Gε / θ θ Z h LQ I: ZL ηL, θ h LQ T I:ε ZLε1; ηLθ U
U
h LQ T >(1−ρ2)1/2σε2]−1φ^>ZLµ :ε ηθ θ @>σ(:ε θ θ @`Iε ε Gε U
ZKHUH
Iε ε
πσε12)−1/2 H[S>−ε12 σε12)], λ
µ :ε ηθ θ αλ > ηε1 ±@µ σε2ρ/σε1)ε1,
σ(:ε θ θ (1−ρ2)1/2 σε2,
DQG θ {α, λ, µ2, ρ, σε1, σε2}. 5HIHU IRU IXUWKHU GHWDLOV WR &KDSWHU 1RWHWKDWVLQFH:LVDUHODWLRQDOPRGHOWKH/3 η LVSDUWRIWKHHUURU GHQVLW\IXQFWLRQ>XQOLNHLQ @ 6XSSRVH QRZ WKDW Q VDPSOH GDWD ZHUH FROOHFWHG ZLWK VRPH RI WKH REVHUYDWLRQV ULJKWFHQVRUHG 7\SH , FHQVRULQJ 'HQRWH E\ /7 WKH ORJ FHQVRULQJ WLPH /7 ORJ 7 DQG OHW ): /7 ηL θ EH WKH FXPXODWLYH GLVWULEXWLRQIXQFWLRQ &') RI:IRUWKHFHQVRUHGREVHUYDWLRQL 7KHQWKHOLNHOLKRRGIXQFWLRQLV / θ θ) h LQ I: ZL ηLθ θ)δL>): /7 ηLθ θ)]1−δL
ZKHUH LIZL LVJLYHQ REVHUYHG δL LIZL LVFHQVRUHG QRQREVHUYDEOH 6HSDUDWLQJ WKH JLYHQ GDWDVHW LQWR WZR VXEVHWV RI N REVHUYHG FRPSOHWH REVHUYDWLRQVDQGNFHQVRUHGREVHUYDWLRQV NN Q WKHORJ OLNHOLKRRGIXQFWLRQ//LVIURP // θ θ ORJ>/ θ θ)]
N
L
Ҟ( δL ORJ>I: ZL ηLθ θ @
N
δ ORJ>) L
θ : /7 ηLθ
L
7KH 0/ HVWLPDWHV IRU WKH SDUDPHWHUV LQ θ DUH WKRVH ZKLFK PD[LPL]H $GHWDLOHGHVWLPDWLRQSURFHGXUHLVQRZGHYHORSHG
5(63216(02'(/,1*0(7+2'2/2*<
)URP WKH H[SUHVVLRQV IRU WKH FRQGLWLRQDO GI RI : WKH PDUJLQDO XQFRQGLWLRQDO H[SUHVVLRQIRUWKHGILV UHIHUWR&KDSWHUV DQG U
I: Z η, θ T >σ(] θ θ ]−1 φ^>Zµ ] η, θ @>σ(θ θ @`φ ] G] U
6LPLODUO\WKHPDUJLQDO&')RI:LV
): Z η,θ θ)
U
T Φ^>Zµ ] η,θθ @>σ(θθ @`φ ])G]
U
ZKHUHφ(.)LVWKHVWDQGDUGQRUPDOGIΦ LVWKHVWDQGDUGQRUPDO&') DQG µ ] η, θ θ αλ > ησε1] λ@µσε2ρ] σ(θ θ σε2 ρ ,QVHUWLQJ DQG LQWR DQH[SUHVVLRQH[SOLFLWLQWKH SDUDPHWHUV LV REWDLQHG 6RPH DSSUR[LPDWLRQV PD\ QRZ EH DSSOLHG WR IDFLOLWDWHQXPHULFDOPD[LPL]DWLRQRI )LUVW WR HOLPLQDWH WKH QHHG IRU LQWHJUDWLRQ LQ I: Z η θ D 7D\ORUEDVHG DSSUR[LPDWH H[SDQVLRQ KDV EHHQ GHYHORSHG LQ &KDSWHU 7KLVDSSUR[LPDWHH[SUHVVLRQLVJLYHQLQ DQGLVUHLQWURGXFHGKHUH IRUFRQYHQLHQFH U
I: Z η, θ T I:ε1 Zε1; η, θ Iε1 ε Gε U
≅>π(1 ρ2)σε22]−1/2 H[S> =I@ [^ σε12[(1−ρ2)1/2σε2]−2[(αηλ−1 + ρσε2/σε1 =I αηλ−2 λ ρ2)1/2σε2=I@` ZKHUH =I =I Z >Z αλ ηλ µ@[ ρ2)σε22]1/2. 6LPLODUO\ZHZLVKWRREWDLQDVLPSOHDSSUR[LPDWHH[SUHVVLRQIRUWKH VWDQGDUG QRUPDO &') ZKLFK ZRXOG HOLPLQDWH WKH QHHG IRU D VLQJOH QXPHULFDOLQWHJUDWLRQLQREWDLQLQJWKH&')RI: $Q DSSUR[LPDWLRQ ZLWK VXFK GHVLUDEOH SURSHUWLHV KDV EHHQ UHFHQWO\ GHYHORSHGLQ6KRUH D Φ(] H[S^ORJ H[S>%>H[S &] @']@`]! ZKHUH% & ' )RU]!WKLVVLPSOH DSSUR[LPDWLRQ KDV D PD[LPXP HUURU RI ± 1RWH WKDW WKH SDUDPHWHUVKDYHEHHQGHWHUPLQHGVRWKDWWKHPD[LPXPDEVROXWHHUURUIRU
&+$37(55(/,$%,/,7<(1*,1((5,1* :,7+&(1625,1*
]! LV PLQLPL]HG )RU ] WKH DEVROXWH HUURU PD\ EH VRPHZKDW ODUJHU PD[LPDOO\DURXQG ,WLVUHFRPPHQGHGIRU] WRUHSODFH]E\] DQGΦ(] E\Φ(] $ PRUHDFFXUDWHDSSUR[LPDWLRQ DQHUURURI WKRXJKVRPHZKDW DOJHEUDLFDOO\PRUHFRPSOH[DSSHDUVLQ&KDSWHU ,QWURGXFLQJIURP LQWR ZHREWDLQIRUWKH&')RI:DW : /7 ): /7 η, θ)
U
T Φ^>/7µ ] ηθθ @>σ(θθ @`φ ])G]
U
U
T
T j ] ) i ] G] j ] ) i ] G] U
≅
T
H[S^ORJ H[S>%>H[S &]) @'])@` φ ])G]
U U
T
^H[S^ORJ H[S>%>H[S &]) @'])@`} φ ])G]
θ @>σ(θ θ @ ]) >/7µ ] ηθ
ZKHUH
DQGµ ] ηθ θ @DQGσ(θ θ DUHJLYHQE\ DQG UHVSHFWLYHO\ ,QWURGXFLQJ IURP IRU ):DQGIURP IRU I: ZH REWDLQ IRU DQDSSUR[LPDWHH[SUHVVLRQWKDWQHHGVWREHPD[LPL]HGZLWKD VLQJOH QXPHULFDO LQWHJUDWLRQ IRU HDFK FHQVRUHG REVHUYDWLRQ WR GHULYH ): /7 ηLθ θ L N :KLOHWKHQHZDSSUR[LPDWHORJOLNHOLKRRGIXQFWLRQVHHPVVLPSOHUWR VROYH>UHODWLYHWRWKHRULJLQDOEDVHGRQ DQG @PD[LPL]LQJ LVVWLOOQRHDV\QXPHULFDOWDVN WKHSDUDPHWHUV YHFWRULQFOXGHVVL[ SDUDPHWHUV ,W LV WKHUHIRUH VXJJHVWHG WKDW VHSDUDWLRQ RI WKH HVWLPDWLRQ SURFHGXUH LQWR WZR VXEURXWLQHV HDFK HVWLPDWLQJ D GLIIHUHQW VXEVHW RI WKH SDUDPHWHUV EH IROORZHG VLPLODUO\ WR WKH HVWLPDWLRQ SURFHGXUH GHYHORSHGIRUXQFHQVRUHGREVHUYDWLRQVLQ&KDSWHU 6SHFLILFDOO\WKH500VWUXFWXUDOSDUDPHWHUVRI6HW,, α,λ,µ2)ZLOO EH HVWLPDWHG E\ ZHLJKWHG QRQOLQHDU OHDVWVTXDUHV 1//6 XVLQJ RQO\ FRPSOHWHREVHUYDWLRQV7KHHUURUSDUDPHWHUVRI6HW,,,^ρ, σε1, σε2), ZLOO EH HVWLPDWHG YLD PD[LPL]LQJ XVLQJ DOO DYDLODEOH REVHUYDWLRQV
5(63216(02'(/,1*0(7+2'2/2*<
LQFOXGLQJFHQVRUHGREVHUYDWLRQV 7KHUHDGHULVUHIHUUHGWR6HFWLRQ ZKHUHWKHVHVROXWLRQSURFHGXUHVDUHH[SRXQGHGLQGHWDLO )RUWKH1//6SURFHGXUHWKHIROORZLQJDSSUR[LPDWHPHDQYDOXHFDQ EHXVHGIRU UHIHUIRUGHWDLOVWR6HFWLRQ
( : ≅ αλ > η λ@ α λ σε1ηλµ
$FFRUGLQJO\WKHIROORZLQJHVWLPDWLQJURXWLQHLVVXJJHVWHG UHFDOOWKDW LW LV DVVXPHG WKDW YDOXHV RI WKH OLQHDU SUHGLFWRU IRU LQGLYLGXDO REVHUYDWLRQVDUHNQRZQ HVWLPDWHG IURP3KDVHRIWKHDQDO\VLV , 8VLQJ WKH UHVLGXDO YDULDQFH IURP WKH OLQHDU UHJUHVVLRQ VWDJH RI 3KDVH DV DQ HVWLPDWH IRU σε12 DSSO\ 1//6 WR PD[LPL]H 6 LQ RUGHUWRILQGLQLWLDOHVWLPDWHVIRU^α, λ, µ2`8VHXQZHLJKWHG1//6IRU WKLVVWHS8VHRQO\FRPSOHWHREVHUYDWLRQV ,, )RU WKH GHULYHG HVWLPDWHV RI ^α, λ, µ2` PD[LPL]H WKH ORJ OLNHOLKRRG IXQFWLRQ WR ILQG HVWLPDWHV IRU WKH HUURU SDUDPHWHUV ^ρ, σε1, σε2}. 8VHERWKFRPSOHWHDQGFHQVRUHGREVHUYDWLRQVҏ ,,, &DOFXODWH YDULDQFHV IRU LQGLYLGXDO FRPSOHWH REVHUYDWLRQV XVLQJ 0HWKRG ,, GHWDLOHG LQ 6HFWLRQ 8VH WKH UHFLSURFDO YDOXHV RI WKH GHULYHGYDULDQFHVDVZHLJKWVIRUWKHQH[WVWHS ,9'HULYHXSGDWHGHVWLPDWHVIRU^α, λ, µ2`XVLQJZHLJKWHG1//6WR PD[LPL]H6 8VHRQO\FRPSOHWHREVHUYDWLRQV 9 7HVW IRU FRQYHUJHQFH RI WKH VROXWLRQ YDULRXV FULWHULD PD\ EH XVHG IRU H[DPSOH VLJQLILFDQW UHGXFWLRQ LQ WKH UHVLGXDO YDULDQFH ,I FRQYHUJHQFHKDVQRW\HWRFFXUUHGJRWRVWHS,,2WKHUZLVHWHUPLQDWHWKH HVWLPDWLRQSURFHGXUH &RPPHQWV ,W LV DOZD\V GHVLUDEOH WR HVWLPDWH DOO WKH SDUDPHWHUV RI θ YLD PD[LPL]DWLRQ RI ZKHUH DOO REVHUYDWLRQV DUH XVHG LQFOXGLQJ FHQVRUHG RQHV ,I WKH LQLWLDO HVWLPDWHV RI WKH SDUDPHWHUV ^α, λ, µ2` GHULYHG LQ VWDJH , DUH XVHG DV VWDUWLQJ YDOXHV IRU D VHDUFK URXWLQH WKDW PD[LPL]HV DQG HVWLPDWHV IRU DOO VL[ SDUDPHWHUV FDQ EH GHULYHG WKHUHRI WKHQ WKHVH 0/ HVWLPDWHV VKRXOG EH FRQVLGHUHG DV WKH ILQDO 0/
&+$37(55(/,$%,/,7<(1*,1((5,1* :,7+&(1625,1*
HVWLPDWHV IRU DOO SDUDPHWHUV RI WKH 500 PRGHO 2WKHUZLVH VRPH LWHUDWLRQVZRXOGEHQHHGHG ,WLVDOZD\VDJRRGLGHDWRUHGXFHWKHQXPEHURISDUDPHWHUVLQWKH PRGHO E\ DVVXPLQJ WKDW FHUWDLQ SDUDPHWHUV KDYH VSHFLILHG YDOXHV )RU H[DPSOHDVVXPLQJWKDWWKHHUURUVDUHXQFRUUHODWHG ρ HOLPLQDWHVRQH SDUDPHWHU WKDW QHHGVHVWLPDWLQJ /LNHZLVH DVVXPLQJ ρ DOORZV µ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
ORJ < β β ORJ 6γ ε6!γ
ZKHUHβDQGβDUHIDWLJXHFXUYHFRHIILFLHQWVγLVWKH UDQGRP IDWLJXH OLPLW RI WKH VSHFLPHQ DQG ε LV WKH HUURU WHUP 30 KDYH DVVXPHG WKDW WKH IDWLJXH OLPLW LV UDQGRP DQG WKH\ FKDUDFWHUL]H WKH UDQGRP YDULDEOH 9 ORJ γ DIWHU VWDQGDUGL]DWLRQ DV IROORZLQJ HLWKHU WKH VWDQGDUGL]HG VPDOOHVW H[WUHPH YDOXH 6HY GLVWULEXWLRQ RU WKH VWDQGDUGL]HG QRUPDO GLVWULEXWLRQ 7KH GLVWULEXWLRQ RI ε LV VLPLODUO\ PRGHOHG DV HLWKHU 6HY RU QRUPDO 30 DSSOLHG WKH 5)/ PRGHO WR WKH ODPLQDWH SDQHO GDWD JLYHQ E\ 6KLPRNDZDDQG+DPDJXFKL 5HIHUWR6HFWLRQLQ30SDSHU 7KH IROORZLQJ H[SRVLWLRQ RI WKH SUREOHP SXUVXHV 30 ZKR ZHUH DOVR NLQG HQRXJKWRSURYLGHXVZLWKWKHGDWD7KHODWWHUFRPHIURPVSHFLPHQV LQ IRXUSRLQW RXWRISODQH EHQGLQJ WHVWV RI FDUERQ HLJKWKDUQHVV VDWLQHSR[\ODPLQDWH)LEHUIUDFWXUHDQGILQDOVSHFLPHQIUDFWXUHRFFXUUHG
5(63216(02'(/,1*0(7+2'2/2*<
VLPXOWDQHRXVO\7KXVIDWLJXHOLIHLVGHILQHGWREHWKHQXPEHURIF\FOHV XQWLO VSHFLPHQ IUDFWXUH 7KH GDWD VHW LQFOXGHV ULJKWFHQVRUHG REVHUYDWLRQV NQRZQ DV ³UXQRXWV´ LQ IDWLJXH OLWHUDWXUH 7KHVH REVHUYDWLRQVZLOOEHWUHDWHGDVPLVVLQJYDOXHVLQWKH1//6SDUWRIWKH HVWLPDWLRQSURFHGXUH 7KHGDWDDUHGLVSOD\HGLQ7DEOH 7KH UHVSRQVH YDULDQFHV ERWKLQWKH RULJLQDO VFDOH < DQG LQ WKHORJ WUDQVIRUPHGVFDOH : DVZHOODVWKHFRHIILFLHQWRIYDULDWLRQ &9 IRU: DUHJLYHQLQWKHERWWRPWKUHHURZV)RUWKHORJWUDQVIRUPHGUHVSRQVHWKH YDULDQFHVHHPVWRVWDELOL]H 8VLQJ WKH 5)/ PRGHO 30 REWDLQHG WKH IROORZLQJ HTXDWLRQ DVVXPLQJQRUPDOGLVWULEXWLRQIRUERWK9DQGε UHIHUWR7DEOHLQ30
ORJ < ORJ 6γ ε6!γ
ZLWKSDUDPHWHUV
( 9 µγ σ 9 σγ σ(ε) σ
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
&+$37(55(/,$%,/,7<(1*,1((5,1* :,7+&(1625,1*
7DEOH'DWDIRUWKHQXPHULFDOH[DPSOH66WUHVV LQ03D </LIHWLPH QXPEHU RIF\FOHVWRVSHFLPHQIUDFWXUHNLORF\FOHV &HQVRUHGREVHUYDWLRQVDUHVWDUUHG6WDWLVWLFV LQWKHERWWRPWKUHHURZVH[FOXGHFHQVRUHGREVHUYDWLRQV: ORJ < &9LVWKHFRHIILFLHQWRIYDULDWLRQ 6
<
6
<
6
<
6
<
6
<
9DU <
(
(
(
(
9DU :
&9 :
OLQHDUSUHGLFWRUDOLQHDUWUDQVIRUPDWLRQRIVWUHVVLVH[SHFWHGWRGHFUHDVH ZLWKLQFUHDVLQJVWUHVV7KLVLPSOLHVWKDWWKHORJIDWLJXHOLIHPRGHOHGE\ ZRXOGKDYHFRHIILFLHQWRIYDULDWLRQWKDWLQFUHDVHVZLWK6 VWUHVV ,QGHHG WKLV SURSHUW\ LV FRPSDWLEOH ZLWK WKH JLYHQ GDWD 2EVHUYLQJ
5(63216(02'(/,1*0(7+2'2/2*<
&9 : LQ7DEOHZHUHDOL]HWKDWLWLQFUHDVHVZLWK6)RUH[DPSOHIRU 6 ZH KDYH &9 : ZKLOH IRU ODUJHU 6 ZH KDYH ODUJHU&9 : 7KH500PRGHOLVWKXVFRPSDWLEOHZLWKERWK UHTXLUHPHQWVH[SRXQGHGE\30 :HVWDUWWKH500HVWLPDWLRQZLWKPRGHOLQJRIWKHOLQHDUSUHGLFWRU 30PRGHOWKHOLQHDUSUHGLFWRUE\>UHIHUWR @
η + ε1 6γ H[S [ H[S 9 [!9
ZKHUH[ ORJ 6 DQG9DIWHUVWDQGDUGL]DWLRQLVHLWKHUVWDQGDUGQRUPDORU VWDQGDUG6HY$OWHUQDWLYHO\XVLQJWKH500PRGHOWKHOLQHDUSUHGLFWRU ZLWKWKHHUURUPD\EHPRGHOHGE\ η + ε1 ββ6ε1 $WWHPSWLQJWRLPSOHPHQW3KDVH,RIWKHHVWLPDWLRQSURFHGXUHEDVHG RQFDQRQLFDOFRUUHODWLRQDQDO\VLV UHIHUWR&KDSWHUIRUGHWDLOV GLGQRW SURYLGHEHWWHUILW LQWKHSXUVXLQJVWHSZLVHOLQHDUUHJUHVVLRQ WKDQVLPSOH DSSOLFDWLRQ RI WKH ODWWHU WR WKH ORJ IDWLJXHOLIH 7KLV FRXOG SHUKDSV EH DQWLFLSDWHG VLQFH FDQRQLFDO FRUUHODWLRQ DQDO\VLV LV LQWHQGHG WR FRUUHODWH WZR VHWV RI YDULDEOHV ZKHUHDV LQ RXU FDVH RQH VHW FRPSULVHV D VLQJOH YDULDEOH QDPHO\ WKH UHJUHVVRU 6 VWUHVV $SSO\LQJ VLPSOH OLQHDU UHJUHVVLRQWR:ZHREWDLQIRUWKH/3WKHLQLWLDOHVWLPDWH
( : (>ORJ < @ η ±6
7KH )UDWLR YDOXH LV S ZLWK DQ DGMXVWHG 5VTXDUHG RI DQG UHVLGXDO VWDQGDUG GHYLDWLRQ RI σε1 7KH DYHUDJH RI WKH DEVROXWH YDOXH RI WKH UHODWLYH HUURU UHODWLYH GHYLDWLRQ RI WKH PRGHO IURPWKHREVHUYHGYDOXH LV ,QWURGXFLQJ IURP IRU η LQWR DQG LPSOHPHQWLQJ XQ ZHLJKWHG 1//6 WR ILQG HVWLPDWHV IRU WKH 500 SDUDPHWHUV ^α, λ, µ2}, LQVHUWLQJIRUσε1WKHYDOXHHVWLPDWHGIURPWKHOLQHDUUHJUHVVLRQ DQDO\VLVZHREWDLQWKHSDUDPHWHUV HVWLPDWHV
α λ µ2
7KH DV\PSWRWLF )UDWLR LV KXJH DQG WKH UHVLGXDO VWDQGDUG GHYLDWLRQ LV 7KH DYHUDJH RI WKH DEVROXWH YDOXH RI WKH UHODWLYHHUURU UHODWLYHGHYLDWLRQRIWKHPRGHOIURPWKHREVHUYHGYDOXH LV )LJXUH GLVSOD\V WKH UHODWLYH HUURUV IRU YDULRXV YDOXHV RI : WKHORJIDWLJXHOLIH
&+$37(55(/,$%,/,7<(1*,1((5,1* :,7+&(1625,1*
)LJXUH5HODWLYHHUURUV LQ IRUPRGHO YV:
&RPSDULQJWKHUHVLGXDOVWDQGDUGGHYLDWLRQVREWDLQHGIURPWKH1//6 ZLWK WKDW REWDLQHG IURP OLQHDU UHJUHVVLRQ YV UHVSHFWLYHO\ LW LV GRXEWIXO WKDW WKH QRQOLQHDU PRGHO DGGV PXFK WR WKH JRRGQHVVRIILWUHODWLYHWRWKHOLQHDUPRGHO)XUWKHUPRUHHVWLPDWHVRI α DQGλDUHYHU\KLJKO\FRUUHODWHG ρ= 2QWKHRWKHUKDQGTXDQWLOH YDOXHV FDOFXODWHG IURP WKH 500 PRGHODUH DSSUHFLDEO\ PRUH DFFXUDWH DV ZLOO VKRUWO\ EH GHPRQVWUDWHG 5HVXOWV IRU ERWK PRGHOV ZLOO EH SUHVHQWHG ,QWURGXFLQJ IURP LQWR DQG PD[LPL]LQJ ZLWK UHVSHFW WR ^σε1, σε2} DVVXPLQJ XQFRUUHODWHG HUURUV QDPHO\ ρ ZH REWDLQ
σε1 σε2
ZLWK PD[LPXP ORJOLNHOLKRRG RI 1RWH WKH SUR[LPLW\ EHWZHHQ WKHYDOXHRIWKHSDUDPHWHUσIRUWKHQRUPDOQRUPDOFDVHLQ30 VPRGHO σ DQGWKHFRUUHVSRQGLQJVWDWLVWLFKHUH σε2= ,WHUDWLQJWKHVROXWLRQSURFHGXUHSDUWLFXODUO\LQRUGHUWRLPSURYHWKH LQLWLDO HVWLPDWH RI WKH OLQHDU SUHGLFWRU GRHV QRW \LHOG D EHWWHU VROXWLRQ )URPWKHOLQHDUPRGHOJLYHQE\ WKHILQDOPRGHOLV
< H[S V =
ZKHUH=LVDVWDQGDUGQRUPDOYDULDEOH
5(63216(02'(/,1*0(7+2'2/2*<
7KHQRQOLQHDUILWWHG500PRGHOLV ZLWKηJLYHQE\ DQGWKH500SDUDPHWHUVJLYHQE\WKHHVWLPDWHVLQ DQG 7DEOH VKRZV TXDQWLOH YDOXHV IRU IDWLJXH OLIH IRU GLIIHUHQW YDOXHV RI VWUHVV DQG GLIIHUHQW PRGHOV 9DOXHV IRU 30 PRGHO WKH 5)/ PRGHO XQGHU YDULRXV GLVWULEXWLRQDO VFHQDULRV DUH WDNHQ IURP 30 7DEOH 9DOXHV IRU WKH ORJ IDWLJXHOLIH OLQHDU PRGHO DUH WDNHQ IURP LQWURGXFLQJWKHUHLQ= 9DOXHVIRUWKHORJIDWLJXH OLIH500PRGHODUHFDOFXODWHGQXPHULFDOO\E\VROYLQJ
): Z η,θ θ)
ZKHUHZSLVWKHSWKTXDQWLOHRI:DQG):LVREWDLQHGQXPHULFDOO\IURP 2QH UHDOL]HV WKDW WKH VLPSOHU OLQHDU PRGHO SURYLGHV TXDQWLOHYDOXHVWKDWDUHH[FHSWLRQDOO\GHYLDQWUHODWLYHWRWKHRWKHUPRGHOV REVHUYHYDOXHVIRU6≥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
6WUHVVOHYHO 6
5)/0 DFFRUGLQJWRWKHGLVWULEXWLRQVRI9DQGRI ORJ < JLYHQγ) 6HY 6HY
1RUPDO 1RUPDO
6HY 1RUPDO
1RUPDO 6HY
500PRGHO /LQHDU 1RQOLQHDU (T (T
&+$37(55(/,$%,/,7<(1*,1((5,1* :,7+&(1625,1*
5HIHUHQFHV >@ 3DVFXDO ) * DQG 0HHNHU : 4 $QDO\VLV RI IDWLJXH GDWD ZLWK UXQRXWV EDVHG RQ D PRGHO ZLWK QRQFRQVWDQW VWDQGDUG GHYLDWLRQ DQG D IDWLJXH OLPLW SDUDPHWHU-RXUQDORI7HVWLQJDQG(YDOXDWLRQ >@3DVFXDO)*DQG0HHNHU:4 (VWLPDWLQJIDWLJXHFXUYHVZLWKWKHUDQGRP IDWLJXHOLPLWPRGHO7HFKQRPHWULFV >@ 3DVFXDO ) * $ VWDQGDUGL]HG IRUP RI WKH 5DQGRP )DWLJXH/LPLW 0RGHO &RPPXQLFDWLRQVLQ6WDWLVWLFV 6LPXODWLRQDQG&RPSXWDWLRQ >@ 3DVFXDO ) * 7KH UDQGRP IDWLJXHOLPLW PRGHO LQ PXOWLIDFWRU H[SHULPHQWV -RXUQDORIVWDWLVWLFDO&RPSXWDWLRQDQG6LPXODWLRQ,QSUHVV >@6KLPRNDZD7DQG+DPDJXFKL< 6WDWLVWLFDOHYDOXDWLRQRIIDWLJXHVWUHQJWK LQ FLUFXODUKROHG QRWFKHG VSHFLPHQV RI D FDUERQ HLJKWKDUQHVVVDWLQHSR[\ ODPLQDWH ,Q 6WDWLVWLFDO 5HVHDUFK RQ )DWLJXH DQG )UDFWXUH FXUUHQW -DSDQHVH 0DWHULDOV 5HVHDUFK 9RO 7DQDND 7 1LVKLMLPD 6 ,FKLNDZD 0 (GV (OVHYLHU /RQGRQ >@6KRUH+ D 5HVSRQVH0RGHOLQJ0HWKRGRORJ\ 500 &XUUHQWGLVWULEXWLRQV WUDQVIRUPDWLRQV DQG DSSUR[LPDWLRQV DV VSHFLDO FDVHV RI WKH 500 HUURU GLVWULEXWLRQ &RPPXQLFDWLRQV LQ 6WDWLVWLFV 7KHRU\ DQG 0HWKRGV >@ 6KRUH + E 7KH 5DQGRP )DWLJXH /LIH 0RGHO DV D VSHFLDO FDVH RI WKH 500 PRGHO$FRPPHQWRQ3DVFXDO D &RPPXQLFDWLRQVLQ6WDWLVWLFV 6LPXODWLRQ DQG&RPSXWDWLRQ
This page intentionally left blank
&KDSWHU
6RIWZDUH5HOLDELOLW\*URZWK0RGHOV
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µ W 0RGHOLQJ VRIWZDUH UHOLDELOLW\JURZWK XVXDOO\ UHIHUV WR WKH PRGHOLQJ RI µ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
5(63216(02'(/,1*0(7+2'2/2*<
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
)LJXUH([DPSOH1XPEHURIFXPXODWLYHGHWHFWHGIDLOXUHVYVQXPEHURIVRIWZDUH WHVWLQJKRXUV 0XVD V0GDWDVHW
&+$37(562)7:$5(5(/,$%,/,7<*52:7+02'(/6
7DEOH([DPSOH0XVD V0GDWDVHW1RLVWKHFXPXODWLYHQXPEHURI IDLOXUHVE\WLPHW
1R
W
1R
W
1R
W
1R
W
5(63216(02'(/,1*0(7+2'2/2*<
UHVSRQVH( < DQGIRUWKHPHDQRIWKHORJWUDQVIRUPHGUHVSRQVH( : DUHDVVXPLQJXQFRUUHODWHGHUURUV UHIHUWR&KDSWHUIRUGHWDLOV : ORJ < αλ > ηε1 λ@µε2 αλ > ησε1= λ@µσε2=
( < ≅H[S> αλ ηλ µ σε22/2)] [^ σε1 >η λ αηλ−2α λ @`
λ
( : (^ αλ > ησε1= @`µ ≅ αλ > η λ@ α λ σε1ηλµ
7KHVHH[SUHVVLRQVZLOOEHXVHGODWHULQWKHDQDO\VLV 7KH500HVWLPDWLRQSURFHGXUHVWDUWVZLWKGHWHUPLQLQJWKHVWUXFWXUH RIWKHOLQHDUSUHGLFWRU /3 &RQVLGHUDFRPELQHGPHDQIXQFWLRQRIERWK FXUUHQWVRIWZDUHUHOLDELOLW\JURZWKPRGHOV,,,DQG9 WKHVHDUHGHILQHGLQ 6HFWLRQ µ W D >ORJ EW @F
ZKHUH ^DEF` DUH SDUDPHWHUV WKDW QHHG WR EH GHWHUPLQHG 1RWH WKDW IRU F ZHREWDLQ0RGHO,,,DQGIRUE ZHREWDLQ0RGHO9 0RGHO LV HDVLO\ UHFRJQL]HG DV WKH PHDQ RI DQ 500 PRGHO $VVXPHWKDWσε12 DQGσε22LQ DUHQHJOLJLEOH VHWWLQJWKHPHTXDOWR 7KHQ LVDVSHFLDOFDVHRI LIZHLQWURGXFHLQWKHODWWHULQ WHUPVRIWKHSDUDPHWHUVRI λ , α Fµ2 ORJ D η ORJ EW
/HW XV UHSODFH E\ WKH PRUH JHQHUDO 500 PRGHO +RZHYHU SUHVHUYHWKH/3 η DVJLYHQLQ )RUWKHUHVSRQVHPHDQVJLYHQE\ DQG QHJOHFWLQJWHUPVWKDWLQFOXGHWKHHUURUV YDULDQFHVZH REWDLQUHVSHFWLYHO\
( < H[S> αλ ηλ µ2@ H[S^ αλ ^>ORJ EW @λ`µ2`
( : (>ORJ < @ αλ ^>ORJ EW @λ`µ2
,QWKHDQDO\VLVWKDWIROORZVZHGRQRWSUHVSHFLI\YDOXHVIRUαλRU µ2 DQG OHW WKH GDWD GHFLGH WKHVH YDOXHV )XUWKHUPRUH VLQFH WKH /3 FRQWDLQVD VLQJOH XQNQRZQ SDUDPHWHU E ZH LQFOXGH HVWLPDWLRQ RI WKLV SDUDPHWHUZLWKLQWKHQRQOLQHDUOHDVWVTXDUHV 1//6 SURFHGXUH7KXV WKHQHHGWRLWHUDWHEHWZHHQ1//6DQGDQRWKHUSURFHGXUH WKDWHVWLPDWHV
&+$37(562)7:$5(5(/,$%,/,7<*52:7+02'(/6
WKH /3 LV DYRLGHG +RZHYHU NHHS LQ PLQG WKH DSSUR[LPDWLQJ DVVXPSWLRQV PDGH KHUH ZKLFK UHQGHU WKH HVWLPDWLRQ SURFHGXUH QXPHULFDOO\ VLPSOHU :H KDYH DVVXPHG WKDW WKH HUURUV DUH XQFRUUHODWHG ρ DQGZHKDYHDVVXPHGWKDWWKHVWDQGDUGGHYLDWLRQVRIWKHHUURUVDUH VPDOO HQRXJK WR EH LJQRUHG LQ PRGHOLQJ WKH PHDQ σε1 σε2 ) 7KHVH DVVXPSWLRQVUHGXFHGWKHQXPEHURISDUDPHWHUVWKDWQHHGHVWLPDWLQJE\ DOORZLQJXVWRXVHRQO\1//6WRHVWLPDWHWKH500SDUDPHWHUV $SSO\LQJ1//6WRWKHJLYHQGDWDVHW ZLWKQ ZHREWDLQIRUWKH SDUDPHWHUVRIWKHPHDQIXQFWLRQ α λ µ2 E
:KLOH WKH TXDOLW\ RI WKH DOOLHG VWDWLVWLFV OLNH VWDQGDUG HUURUV RI WKH HVWLPDWHV RU FRUUHODWLRQV EHWZHHQ HVWLPDWHV LV OHVV WKDQ GHVLUDEOH WKH SDUDPHWHUV HVWLPDWHV LPSO\ WKDW WKH IROORZLQJ PD\ FRQVWLWXWH D JRRG PRGHOIRUWKHGDWD ( : (>ORJ < @ >ORJ EW @ QDPHO\α= λ µ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
5(63216(02'(/,1*0(7+2'2/2*<
)LJXUH([DPSOH3UHGLFWHGYVDFWXDOQXPEHURIIDLOXUHV ERWKRQDORJVFDOH
)LJXUH([DPSOH(UURUV DFWXDOPLQXVPRGHO IRUWKHQXPEHURIIDLOXUHVYVWKH DFWXDOQXPEHURIIDLOXUHV ERWKRQDORJVFDOH
&+$37(562)7:$5(5(/,$%,/,7<*52:7+02'(/6
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` ZKHUH QL LV WKH QXPEHU RI IDLOXUHV GXULQJ WLPH LQWHUYDO >WL WL WKH ORJ OLNHOLKRRG IXQFWLRQ // DVVXPLQJDQXQGHUO\LQJ3RLVVRQODZLV N
// ORJ> h I>QL_µ WL µ WL @ L
N
N
N
L
L
L
>µ WL µ WL @ QLORJ>µ WL µ WL @ ORJ QL N
N
L
L
µ WN QLORJ>µ WL µ WL @ ORJ QL
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH([DPSOH0XVD V0GDWDVHW 1R
W
1R
W
)RUDJLYHQPRGHOIRUWKHUHVSRQVHPHDQµ W WKHSDUDPHWHUV YDOXHV WKDWPD[LPL]H DUH0/(V 8VLQJ0XVD V0GDWDVHW0/(VIRUWKHSDUDPHWHUV^DE`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
&+$37(562)7:$5(5(/,$%,/,7<*52:7+02'(/6
7DEOH([DPSOH0/(VDQGRWKHUUHVXOWVIURPWKHDQDO\VLVRI0XVD V0GDWDVHW 0RGHO , ,, ,,, ,9 9
D
E
//
&RQVLGHUWKHVLPSOHVWPRGHOIRU/3 η W W
67'UHV
)URPDWKHRUHWLFDOSRLQWRIYLHZWKLVLVWKHSUHIHUUHGFKRLFHVLQFHWKLV /3 HPERGLHV WKH LGHDO RI WKH QHZ DSSURDFK QDPHO\ VLPSOLFLW\ LQ WKH GHILQLWLRQ RI WKH /3 2QO\ LI WKLV FKRLFH GRHV QRW SURYLGH UHVXOWV FRPSHWLWLYHZLWKWKRVHREWDLQHGE\WKHFXUUHQWPRGHOV 0RGHOV,9 ZLOO ZHRSWIRUDQH[SDQGHG/3ZKLFKZLOOEHDOLQHDUWUDQVIRUPDWLRQRIWKH η W DERYHQDPHO\ η W ββW
D
(TXDWLRQ D PD\ EH DQ RSWLRQDO PRGHO RQO\ LIWKH SDUDPHWHUV HVWLPDWHV DUH VLJQLILFDQWO\ GLIIHUHQW IURP ]HUR 7KHUHIRUH ZH VWDUW WKH PRGHOLQJSURFHVVE\H[DPLQLQJWKHYLDELOLW\RI D )URP DQG D WKHUHVSRQVHTXDQWLOHLV
: αλ)> ββW ε1 λ±@µε2
,QWURGXFLQJIRUε2 LWVH[SHFWHGYDOXH( ε2 ZHREWDLQIRUWKHLWK REVHUYDWLRQWKHWUDQVIRUPHGUHVSRQVHYDOXH
η WL ε1 ββWLε1 > λ /α)(ZLµ @ λ
ZKHUHε1LVWKHUDQGRPHUURUDVVRFLDWHGZLWKWKHLWKYDOXHRIWKH/3DQG ZLLVWKHLWKORJJHGYDOXHRIWKHRULJLQDOUHVSRQVH 7KH GDWD WUDQVIRUPDWLRQ JLYHQ E\ WKH 5+6 RI PD\ VHUYH DV UHVSRQVH LQSXW IRU OLQHDU UHJUHVVLRQ DQDO\VLV RQFH LQLWLDO YDOXHV IRU WKH 500SDUDPHWHUV α, λ, µ2), KDYHEHHQREWDLQHGYLD1//6 :HVWDUWWKHILUVWLWHUDWLRQDVVXPLQJσε1 LQWKHH[SUHVVLRQIRU( : ,QVXEVHTXHQWLWHUDWLRQVDQHVWLPDWHRIσε1ZLOOEHDYDLODEOHIURP WKHPRVWUHFHQWDSSOLFDWLRQRIWKHOLQHDUUHJUHVVLRQVWDJHRIWKHDQDO\VLV DQGWKLVHVWLPDWHPD\EHXVHGLQWKHH[SUHVVLRQIRU( : .
5(63216(02'(/,1*0(7+2'2/2*<
)LJXUH([DPSOH6FDWWHUSORWVRIUHVLGXDOV YVH[DFWYDOXHRI< IURPILWWLQJ WRS WRERWWRP 0RGHO,90RGHO9DQGWKH500PRGHO> DQG @
&+$37(562)7:$5(5(/,$%,/,7<*52:7+02'(/6
$OVRQRZHLJKWLQJLVLQWURGXFHGLQWKHILUVWLWHUDWLRQ ,QVXEVHTXHQW LWHUDWLRQVZHPD\XVHZHLJKWLQJLIWKHUHVLGXDOYDULDQFHYDULHVIURPRQH REVHUYDWLRQ WR WKH QH[W WKDW LV σε1 LV QRW QHJOLJLEOH UHODWLYH WR W\SLFDO YDOXHVRIη $ILUVWUXQRI1//6XVLQJWKHDSSUR[LPDWHSURFHGXUHGHOLYHUV α λ µ 67'UHV 7KHODWWHUVWDWLVWLFLVDERXWRIWKHEHVWRIWKH67'UHVLQ7DEOH 2EVHUYLQJ WKH UHVLGXDOV ZH ILQG REVHUYDWLRQ WR KDYH DQ RXWO\LQJ UHVLGXDO RI ([FOXGLQJ WKLV REVHUYDWLRQ IURP WKH DQDO\VLV ZH REWDLQ α λ µ 67'UHV
7KHODWWHUVWDWLVWLFLVDERXWRIWKHEHVWRIWKH67'UHVLQ7DEOH $SSUR[LPDWH FRQILGHQFH LQWHUYDOV IRU WKH HVWLPDWHV RI WKH 500 SDUDPHWHUV PD\ EH GHULYHG IURP WKH 1//6 SURFHGXUH DV WKHVH DUH SURYLGHG DV VWDQGDUG RXWSXW IURP DQ\ VWDWLVWLFDO SDFNDJH WKDW SHUIRUPV 1//6DQDO\VLV 7XUQLQJ QH[WWR WKH OLQHDUUHJUHVVLRQ SDUW RIWKH DQDO\VLV ZH PRGHO WKH/3DVLQ ZLWKWKHDERYH500SDUDPHWHUVWRREWDLQ β β *RRGQHVVRIILW VWDWLVWLFV JLYH DQ ) YDOXH RI 5DGM RI DQGσε1 %RWKHVWLPDWHVRIβDQGβDUHVLJQLILFDQWLPSO\LQJWKDW WKH\ DUH QRW ]HUR 0RGHO D LV WKHUHIRUH D SRVVLEOH FDQGLGDWH IRU PRGHOLQJWKH/3 &RQVLGHU WKH HVWLPDWH IRU β0. 6LQFH W LV PHDVXUHG RQ D VFDOH RI QDPHO\β0 W QRWPXFKDFFXUDF\LVVDFULILFHGLIZHVHOHFWβ β >DV LQLWLDOO\ VXJJHVWHG UHIHU WR @ :LWK WKHVH YDOXHV WKH QHZ HVWLPDWH LV σε1 5HUXQQLQJ 1//6 WKH ILQDO HVWLPDWHV IRU WKH 500PRGHODUHQHDUO\WKHVDPHDVLQ $VFDWWHUSORWRIWKHUHVLGXDOVIURPWKLVPRGHOLVJLYHQLQ)LJXUH ERWWRPSORW 7KHVFDWWHURIWKHUHVLGXDOVIRUWKHQHZPRGHOVHHPVPRUH UDQGRP WKDQ GLVSOD\HG E\ WKH RWKHU WZR PRGHOV QR IRUPDO VWDWLVWLFDO DQDO\VLV RI UDQGRPQHVV KDV EHHQ SHUIRUPHG )XUWKHUPRUH ZKLOH DSSOLFDWLRQRIWKH500PRGHOUHTXLUHGHVWLPDWLRQRIWKUHHSDUDPHWHUV ^α λ, µ2}, YV WZR SDUDPHWHUV IRU WKH RWKHU WZR PRGHOV WKH HUURU DV
5(63216(02'(/,1*0(7+2'2/2*<
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
&KDSWHU
0RGHOLQJD&KHPR5HVSRQVH
,QWURGXFWLRQ 0RQRWRQH FRQYH[FRQFDYH UHODWLRQVKLSV DERXQG LQ DOO EUDQFKHV RI FKHPLVWU\DQGFKHPLFDOHQJLQHHULQJ7KLVKDVEHHQGHPRQVWUDWHGLQWKH VXUYH\ RI &KDSWHU ZKHUH FXUUHQW PRGHOV UHODWHGWR FKHPRUHVSRQVHV OLNH OLTXLG GHQVLW\ OLTXLG KHDW FDSDFLW\ RU $UUKHQLXV HTXDWLRQ ZHUH UHYLHZHG,Q&KDSWHUZHUHDGGUHVVHGWKHVHPRGHOVDQGKDYHVKRZQ WKDW WKH\ PD\ EH FRQVLGHUHG VSHFLDO FDVHV RI WKH 500 PRGHO ,Q WKLV FKDSWHU 500 LV DSSOLHG WR PRGHO UHDO GDWDVHWV LQ WKH FKHPLFDO HQJLQHHULQJ DUHQD DQG W\SLFDO JRRGQHVVRIILW REWDLQDEOH LQ VXFK PRGHOLQJHQGHDYRUVLVGHPRQVWUDWHG )RUPRQRWRQHFRQYH[FRQFDYHUHODWLRQVKLSVLWLVDFFHSWDEOHSUDFWLFH ZLWKLQ WKH FKHPLFDOHQJLQHHULQJ GLVFLSOLQH WR HQKDQFH WKH JRRGQHVVRI ILW RI HPSLULFDO PRGHOV E\ XVLQJ KLJKRUGHU SRO\QRPLDO PRGHOV )RU H[DPSOH WKH EDVLF $QWRLQH HTXDWLRQ ZKLFK UHODWHV YDSRU SUHVVXUH WR WHPSHUDWXUHLVGHULYHGIURPWKHRUHWLFDODUJXPHQWV UHIHUWR&KDSWHUIRU GHWDLOV 7KLV UHODWLRQVKLS ZDV HQKDQFHG E\ ERWK :DJQHU DQG 5LHGHO 6HFWLRQ ZKR KDG DGGHG SRO\QRPLDO WHUPV WKDW UHVXOWHG LQ HPSLULFDO PRGHOV ZLWK PRUH SDUDPHWHUV WKDQ LQ $QWRLQH HTXDWLRQ 7KH HQKDQFHG PRGHOV KDYH EHHQ VKRZQ WR DWWDLQ EHWWHU JRRGQHVVRIILW 'DXEHUW +LJKRUGHUSRO\QRPLDOVKRZHYHUPD\UHVXOWLQXQVWDEOHPRGHOV$V GLVFXVVHGLQ6KRUH%UDXQHUDQG6KDFKDP WKLVPD\RFFXUZKHQ WKHOHQJWKVRIFRQILGHQFHLQWHUYDOVIRUWKHPRGHO VSDUDPHWHUVDUHODUJHU WKDQWKHSDUDPHWHUV HVWLPDWHV)XUWKHUPRUHKLJKRUGHUSRO\QRPLDOVWHQG
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(502'(/,1*$&+(025(63216(
7R ILW WKH 500 PRGHO ZH DVVXPH WKDW ORJ 3 KDV DSSUR[LPDWHO\ D FRQVWDQW YDULDQFH 5HFDOO WKDW IRU WKH UHVSRQVH < DQG WKH ORJ WUDQVIRUPHG UHVSRQVH : ORJ < ZH KDYH ZLWK WKH DVVRFLDWHG DSSUR[LPDWHH[SUHVVLRQVIRUWKHPHDQV UHIHUWR&KDSWHU
: αλ > ηε1 λ@µε2
αλ > ησε1= λ@µσε2=
λ
2
( < ≅H[S> αλ η µ σε2 )]
[^ σε1 >η λ αηλ−2α λ @`
( : (^ αλ > ησε1= λ@`µ
≅ αλ > η λ@ α λ σε1ηλµ
ZKHUH ε1 σε1= DQG ε σε2= DUH DVVXPHG WR EH XQFRUUHODWHG QRUPDO HUURUV = DQG = DUH LQGHSHQGHQW VWDQGDUG QRUPDO YDULDEOHV DQG θ= θ {α λ µ2σε1 σε2`LVWKHSDUDPHWHUV YHFWRU $VVXPHWKDW7 µ7ε1ZKHUHµ7LVWKHH[SHFWHGYDOXHRI7)RUPRVW UHSRUWHGPHDVXUHPHQWVRI7LWLVUHDVRQDEOHWRDVVXPHWKDW_ε1_/µ7 << 1. ,I WKDW LV YDOLG ZH FDQ QHJOHFW ε σε1≅ DQG DVVXPH WKDW WKH HUURU YDULDQFHRI: ORJ 3 LVGHWHUPLQHGVROHO\E\ ε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
280
RESPONSE MODELING METHODOLOGY
Figure 17.1. Plot of the Oxygen data (pressure vs. temperature)
Introducing σε1=0 in E[log(P)] (17.3), we obtain, via NL-LS estimation, initial values of α = 603.1, λ = -0.9300, µ2 = -640.9. The estimated value of λ is near the value given by Antoine equation (λ=-1). From (17.1), recalling that E(ε2)= 0, we obtain explicitly for the linear predictor (LP) of observation i ηi + ε1 = {[log(Pi)+640.9](-0.93/603.1)+1}(-1/0.93),
(17.4)
where log(Pi) is an estimate of E(W)=E[log(Pi)] for observation i. Note that assuming uncorrelated errors and that σε1 is negligible allowed derivation of (17.4). The latter provides a non-linear response transformation from which the coefficients of the LP can be estimated straightforwardly, via stepwise linear regression. This "spares" us the more complex "standard" estimation procedure, expounded in Chapter 8.
&+$37(502'(/,1*$&+(025(63216(
7R HVWLPDWH WKH OLQHDU SUHGLFWRU /3 FRQVLGHU DQ LQLWLDO PRGHO IRU REVHUYDWLRQL ηLε1 ββµ7Lε1, ZKHUHµ7LLVWKHH[SHFWHGYDOXHRIWHPSHUDWXUHUHDGLQJIRUREVHUYDWLRQL $SSO\LQJ OLQHDU UHJUHVVLRQ WR WKH WUDQVIRUPHG UHVSRQVH YDOXHV ZH REWDLQ η 7 ZKHUH ZH UHSODFH µ7L E\ 7L WKH DFWXDO WHPSHUDWXUH UHDGLQJ IRU REVHUYDWLRQL7KHDVVRFLDWHG)5DWLRLV(5DGMLV DQGWKH VWDQGDUGGHYLDWLRQRIWKHUHVLGXDOV σε1 LV :KLOHWKHHVWLPDWHRI β0 LVVLJQLILFDQWO\QRQ]HUR W S WKHHVWLPDWHRIβKDVDQ DSSUHFLDEO\ ODUJHU W YDOXH RI 7KLV LQGHHG RYHUVKDGRZV WKH VLJQLILFDQFHRIβ)XUWKHUPRUHWKHODWWHULVQHJOLJLEOHUHODWLYHWRWKHVL]H RI7 7KH SUREOHP ZLWK WKH VLJQLILFDQW EXW VPDOO LQWHUFHSW HVWLPDWH HQFRXQWHUHGLQ([DPSOHRI6HFWLRQLVUHSHDWHGKHUHDQGWKHVDPH GHWHUPLQDWLRQ LV PDGH WR VHW β 5HUXQQLQJ OLQHDU UHJUHVVLRQ WKH UHVXOWVREWDLQHGDUHβ ZLWK)5DWLR (5DGM DQGσε1 7KHVPDOOYDOXHRIσε1 (UHODWLYHWR7ZKLFKLVPHDVXUHGLQ VKRZV WKDW WKH UDQGRP FRPSRQHQW GHOLYHUHG WR WKH UHVSRQVH YLD 7 LV LQGHHG VPDOO UHODWLYH WR µ7 DV LQLWLDOO\ SRVWXODWHG ,PSOHPHQWLQJ WKH 1//6 URXWLQH ZLWK QR ZHLJKWLQJ >QDPHO\ DVVXPLQJ D FRQVWDQW YDULDQFH IRU ORJ 3 @LVWKHUHIRUHVXSSRUWHGE\WKHDQDO\VLVUHVXOWV 2EVHUYLQJVWXGHQWL]HGUHVLGXDOVZHQRWHWKDWVRPHRIWKHXSSHUVFDOH REVHUYDWLRQV DUH RXWOLHUV DEVROXWH YDOXH RI VWXGHQWL]HG UHVLGXDO ODUJHU WKDQ 'HOHWLQJ REVHUYDWLRQV ZLWK RXWO\LQJ GHYLDWLRQV REVHUYDWLRQV KDG QRW PHDQLQJIXOO\ LPSURYHG WKH ILW DQG WKHUHIRUH WKHVH REVHUYDWLRQVZHUHOHIWLQWKHDQDO\VLV7KHILQDOPRGHOLV
ORJ 3 > µ7ε1 @ ε2
2QFH DOO SDUDPHWHUV KDYH EHHQ HVWLPDWHG H[SHFWHG YDOXHV IRU LQGLYLGXDO REVHUYDWLRQV PD\ EH FDOFXODWHG DV ZHOO DV DQ\ GHVLUDEOH TXDQWLOH 7RDVVHVVWKHJRRGQHVVRIILWRIWKHHVWLPDWHG500PRGHO ZH
5(63216(02'(/,1*0(7+2'2/2*<
HYDOXDWHG (>ORJ 3 @ IRU LQGLYLGXDO REVHUYDWLRQV QHJOHFWLQJ WKH WHUPWKDWFRQWDLQVWKHQHJOLJLEOHσε. 5HVLGXDOVIURPWKLVH[SUHVVLRQDUH XVHG WR FDOFXODWH 67' 5HV WKH UHVLGXDO VWDQGDUG GHYLDWLRQ ZKLFK LV GLVSOD\HGLQ7DEOH7KLVWDEOHDOVRVKRZVUHVXOWVIURPWKHDQDO\VHV RIWKHRWKHUGDWDVHWVUHODWLQJWRWKHRWKHUYDSRUV7KHVHDQDO\VHVZHUH FRQGXFWHGVLPLODUO\WRWKHGHWDLOHGDQDO\VLVRI2[\JHQ ,QVSHFWLQJ 7DEOH LW LV ZRUWK QRWLQJ WKDW WKH )UDWLR YDOXHV REWDLQHGIRUDOOYDSRUVDUHDOOPHDVXUHGLQRYHU(7KLVDWWHVWVWRWKH JRRGILWREWDLQHGGXULQJWKHOLQHDUUHJUHVVLRQSKDVHRIWKHDQDO\VLV$OVR WKH YDOXHV RI λ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
/LQHDU5HJUHVVLRQ5HVXOWV )5DWLR ([FOXGHG2EV σε1
$UJRQ ( Q 1LWURJHQ ( Q 2[\JHQ ( Q +HSW ( Q 3URSW ( Q
α
1//65HVXOWV λ µ2
67' 5HV
QRQH
QRQH
QRQH
1R
1R
&+$37(502'(/,1*$&+(025(63216(
([WHQGHG5LHGHO ORJ 3 $%7&ORJ 7 '7(7
:DJQHUHTXDWLRQ ORJ 3U 7U $W%W&W'W
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
2[\JHQ
$UJRQ
1LWURJHQ
(
(
(
(
(
(
(
(
(
(
(
(
5(63216(02'(/,1*0(7+2'2/2*<
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
/LQHDU5HJU5HVXOWV )5DWLR ( ( ( ( ( ( ( (
([FOXGHG 2EV
1//65HVXOWV
67'5HV
σε1
α
λ
µ2
&+$37(502'(/,1*$&+(025(63216(
6LQFH LW LV FXVWRPDU\ LQ PRGHOLQJ &S WR XVH WKH UHVSRQVH LQ WKH RULJLQDO VFDOH 7DEOH LQFOXGHV DOVR UHVXOWV RI ILWWLQJ KHDW FDSDFLW\ GDWDLQWKHRULJLQDOVFDOH &S 7KHPRGHOIRU&SLVWKHDSSUR[LPDWHPHDQ RI WKH UHVSRQVH ( < JLYHQ LQ ZLWK σε1 1RWH WKDW WKH DSSUR[LPDWH DVVXPSWLRQ PDGH HDUOLHU LQ ILWWLQJ ORJ < QDPHO\ WKDW LI σε1LVVPDOOQRZHLJKWLQJLVUHTXLUHGLQWKH1//6SDUWRIWKHSURFHGXUH PD\EHFRPHTXHVWLRQDEOHZKHQ&SLVXVHGDVWKHUHVSRQVH7KHUHIRUHDV D JHQHUDO UXOH LW LV UHFRPPHQGHG WKDW ILWWLQJ ZLOO EH DSSOLHG WR ORJ < UDWKHUWKDQ< $ FRPSDUDWLYH HYDOXDWLRQ LV QRZ FRQGXFWHG :H DGRSW 'DXEHUW¶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
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH([DPSOH&RPSDUDWLYHHYDOXDWLRQRIWKHVWDQGDUGGHYLDWLRQVRIUHVLGXDOV 6XEVWDQFH Q
6WDQGDUG'HYLDWLRQRIUHVLGXDOV IURPPRGHOLQJ&SDV 500
/LQHDU
QGRUGHU SRO\QRPLDO
UGRUGHU SRO\QRPLDO
$FHWRQLWULOH Q $PPRQLD Q %XWDQRLFDFLG Q (WKDQRLFDFLG Q
12 10
{Heat Capacity, CPAmon, vs. temperature}
8
6
4
2
50
100
Cp Actual
Cp calc.
150
)LJXUH3ORWRIH[DFWDQGFDOFXODWHGVROLG&SIRU$PPRQLD
&+$37(502'(/,1*$&+(025(63216(
Heat Capacity, CPButan, vs temperature
100 80 Cp 60 40 )LJXUH 3ORWRIH[DFWDQGFDOFXODWHGVROLG&SIRU%XWDQRLFDFLG Cp 20 120
50
100
150
200
Actual
calc.
)LJXUH3ORWRIH[DFWDQGFDOFXODWHGVROLG&SIRU%XWDQRLFDFLG
$SSO\LQJ500WRD&KHPR5HVSRQVH6HFRQG9DULDWLRQ $YDULDWLRQRIWKH500PRGHOLQWURGXFHGLQ&KDSWHU LV < 0 H[S^ αλ >ηλH[S λε1/η @ε2`
)RUWKHGDWDDGGUHVVHGLQWKLVVHFWLRQZHXVH IRUPRGHOLQJ $VVXPLQJWKDWWKHWZRQRUPDOHUURUVε1 DQGε2,KDYH]HURPHDQDQG VWDQGDUGGHYLDWLRQVσε1 DQGσε2, UHVSHFWLYHO\DQGWKDWWKH\GHULYHIURPD ELYDULDWHQRUPDOGLVWULEXWLRQZLWKFRUUHODWLRQρRQHFDQZULWH UHIHUWR 6HFWLRQ
ε1 σε1=ε2 σε2[ρ= ρ =@
ZKHUH = DQG = DUH WZR LQGHSHQGHQW VWDQGDUG QRUPDO YDULDEOHV ,QWURGXFLQJ IRU ε2 EDFN LQWR DQG DVVXPLQJ η ρ WKXV LQ IDFWREWDLQLQJDQLQYHUVHQRUPDOL]LQJWUDQVIRUPDWLRQ,17 ZHREWDLQ < 0 H[S^ αλ >H[S λσε1= @(σε2ρ) =` ZKHUH ZH UHSODFHG = E\ = 1RWH WKDW WKH FRHIILFLHQWV RI = FDQ KDYH GLIIHUHQWVLJQV ZKHQρ=−1 DSRLQWWKDWZHKDYHUHODWHGWRHDUOLHU $OWKRXJK LVDPRGHORIUDQGRPYDULDWLRQ VLQFHη ZHXVH LWKHUHIRUPRGHOLQJV\VWHPDWLFYDULDWLRQE\WUHDWLQJ=DVWKHLQGHSHQGHQW YDULDEOHH[SUHVVHGLQVWDQGDUGXQLWVGHILQHGE\
= ;0[ σ[
5(63216(02'(/,1*0(7+2'2/2*<
ZKHUH0[DQGσ[UHVSHFWLYHO\DUHWKHPHGLDQDQGWKHVWDQGDUGGHYLDWLRQ RIWKHLQGHSHQGHQWYDULDEOH;(TXDWLRQ WKXVEHFRPH < H[S^$%>H[S &= @'=` ZKHUH^$%&'`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
&+$37(502'(/,1*$&+(025(63216(
7DEOH([DPSOH'DWDFKDUDFWHUL]DWLRQDQGILWRI YDSRUSUHVVXUHGDWD 3DUDPHWHU $ % & ' 9DULDQFH 6DPSOHVL]H Q 7XQLWV 7UDQJH 70HGLDQ 3UHVVXUHXQLWV 3UHVVXUHUDQJH 0HGLDQOQ 3
2[\JHQ 9DOXH &RQILGHQFH ,QWHUYDO ( . 0SD
(
$UJRQ 9DOXH &RQILGHQFH ,QWHUYDO ( . EDU
(
1LWURJHQ 9DOXH &RQILGHQFH ,QWHUYDO ( . EDU
(
7KHUHVXOWVRIWKLVFRPSDULVRQDUHVKRZQLQ7DEOH,WFDQEHVHHQ WKDWWKHDFFXUDF\RI,17 LVRUGHUVRIPDJQLWXGHKLJKHUWKDQWKDW RI WKH SDUDPHWHU $QWRLQH HTXDWLRQ DQG RQO\ VOLJKWO\ ORZHU WKDQ WKH DFFXUDF\ RI WKH SDUDPHWHU H[WHQGHG 5LHGHO HTXDWLRQ :DJQHU¶V HTXDWLRQSURYLGHVWKHPRVWDFFXUDWHFRUUHODWLRQVIRU2[\JHQ$UJRQDQG 1LWURJHQ +RZHYHU WKLV HTXDWLRQ ZDV DFWXDOO\ ILWWHG E\ VWHSZLVH UHJUHVVLRQ IRUWKHYHU\VDPHVXEVWDQFHVDVGHWDLOHGHDUOLHU 7DEOH([DPSOH&RPSDULVRQRIYDSRUSUHVVXUHFRUUHODWLRQHTXDWLRQV
&RUUHODWLRQHTXDWLRQ 1RRISDUDPHWHUV
2[\JHQ
$QWRLQH ([WHQGHG5LHGHO V :DJQHU V ,17
( ( ( (
9DULDQFHDWRSWLPDOVROXWLRQ $UJRQ 1LWURJHQ ( ( ( (
( ( ( (
5(63216(02'(/,1*0(7+2'2/2*<
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
$PPRQLD VROLG 9DOXH &RQILGHQFH ,QWHUYDO
6DPSOHVL]H Q . 7XQLWV 7UDQJH 70HGLDQ FDO.PRO +&XQLWV +&UDQJH 0HGLDQ&S
( ( ( (
$FHWRQLWULOH VROLG %XWDQRLFDFLG OLTXLG 9DOXH &RQILGHQFH 9DOXH &RQILGHQFH ,QWHUYDO ,QWHUYDO . FDO.PRO
(
. -.PRO
(
CHAPTER 17 MODELING A CHEMO-RESPONSE
291
Figure 17.4. Experimental data and calculated curve [using (17.12)] for solid Ammonia’s heat capacity
Figure 17.5. Experimental data and calculated curve [using (17.12)] for liquid Butanoic acid’s heat capacity
5(63216(02'(/,1*0(7+2'2/2*<
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
WKRUGHUSRO\QRP WKRUGHUSRO\QRP UGRUGHUSRO\QRP (T QGRUGHUSRO\QRP
9DULDQFHDWRSWLPDOVROXWLRQ UHODWLYHYDULDQFH $PPRQLD $FHWRQLWULOH %XWDQRLFDFLG
XQVWDEOH
XQVWDEOH XQVWDEOH XQVWDEOH
&+$37(502'(/,1*$&+(025(63216(
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
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH 6KDSHVRIFXUYHVUHSUHVHQWLQJWHPSHUDWXUHGHSHQGHQWSURSHUWLHV 1R
3URSHUW\
&KDQJHZLWKLQFUHDVLQJ WHPSHUDWXUH
6KDSHRIWKHFXUYH
9DSRUSUHVVXUHOLTXLG +HDWFDSDFLW\VROLG +HDWFDSDFLW\OLTXLG 9LVFRVLW\JDV 9LVFRVLW\OLTXLG 6XUIDFHWHQVLRQOLTXLG
LQFUHDVLQJ LQFUHDVLQJ LQFUHDVLQJ LQFUHDVLQJ GHFUHDVLQJ GHFUHDVLQJ
KLJKO\FRQYH[ PRGHUDWHO\FRQFDYH VOLJKWO\FRQYH[ PRGHUDWHO\FRQFDYH KLJKO\FRQFDYH VOLJKWO\FRQFDYH
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
&+$37(502'(/,1*$&+(025(63216(
>@:DJQHU: 1HZYDSRUSUHVVXUHPHDVXUHPHQWVIRUDUJRQDQGQLWURJHQDQGD QHZ PHWKRG IRU HVWDEOLVKLQJ UDWLRQDO YDSRU SUHVVXUH HTXDWLRQV &U\RJHQLFV >@:DJQHU:(ZHUV-3HQWHUPDQQ: 1HZYDSRUSUHVVXUHPHDVXUHPHQWV DQG D QHZ UDWLRQDO YDSRU SUHVVXUH HTXDWLRQ - &KHP 7KHUPRG\QDPLFV
This page intentionally left blank
&KDSWHU
)RUHFDVWLQJ66KDSHG'LIIXVLRQ3URFHVVHV
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
5(63216(02'(/,1*0(7+2'2/2*<
VSHFLILHG WLPH 7 &RQYHUVHO\ ZH PD\ ZLVK WR IRUHFDVW 7 JLYHQ 3 ,Q ERWK FDVHV UHODWLRQDO PRGHOLQJ LV QHHGHG %\ FRQWUDVW WKH ,17V KDYH EHHQRULJLQDOO\GHULYHGDVPRGHOVRIUDQGRPYDULDWLRQ7KLVLVZK\WKH\ DUH GHYHORSHG LQ 3DUW ,9 RI WKH ERRN GHGLFDWHG WR WKH PRGHOLQJ RI UDQGRP YDULDWLRQ ,Q WKLV FKDSWHU WKH ,17V DUH DGDSWHG WR EH XVHG IRU UHODWLRQDOPRGHOLQJDQGWKHQDWXUDOSODFHIRUWKLVDSSOLFDWLRQLV3DUW,,, RI WKH ERRN WR ZKLFK WKLV FKDSWHU EHORQJV $ VLPLODU DGDSWDWLRQ ZDV GRQHLQWKHSUHYLRXVFKDSWHU 6HFWLRQ ZKHUHDQ ,17ZDVXVHGWR PRGHOFKHPLFDOGDWD ,W PD\ EH EHQHILFLDO DW WKLV SRLQW WR WXUQ WR &KDSWHU DQG JHW DFTXDLQWHGZLWKWKH,17VXVHGKHUHIRUPRGHOLQJ6VKDSHGSURFHVVHV 7KHRUHWLFDO%DFNJURXQGIRU6VKDSHG'LIIXVLRQ3URFHVVHV 'LIIXVLRQ SURFHVVHV DUH VWRFKDVWLF SURFHVVHV WKDW GHVFULEH WKH WLPH GHSHQGHQWHYROXWLRQRIWKHGLIIXVLRQRIDUDQGRPSKHQRPHQRQVXFKDV WKHSHQHWUDWLRQLQWRWKHPDUNHWRIDQHZSURGXFWRUWKHDGRSWLRQUDWHE\ WKH UHOHYDQW SRSXODWLRQ RI D WHFKQRORJLFDO LQQRYDWLRQ 0RVW RIWHQ WKH UHVSRQVHRILQWHUHVWLVH[SUHVVHGDVWKHWLPHGHSHQGHQWSHUFHQWDJHRIWKH SRSXODWLRQZKHUHWKHGLIIXVLRQKDVDOUHDG\WDNHQSODFH)RUH[DPSOHWKH SURSRUWLRQRIWKHSRSXODWLRQWKDWKDVDOUHDG\SXUFKDVHGWKHQHZSURGXFW E\ WLPH 7 7KH IRUHFDVWLQJ SUREOHP LV WKDW RI IRUHFDVWLQJ DW WLPH 7 WKHYDOXHRIWKHUHVSRQVHDWWLPH7/JLYHQWKHDGRSWLRQKLVWRU\XSWR DQG LQFOXGLQJ 7 0RUH IRUPDOO\ GHQRWH E\ SW WKH FXPXODWLYH GLIIXVLRQ UDWH E\ WLPH W7 ≤SW≤ 7KHQ WKH UHTXLUHG IRUHFDVW LV WKDW RI ( 37/_SS«S7 2FFDVLRQDOO\LQYHUVHIRUHFDVWVDUHDOVRQHHGHGQDPHO\IRUHFDVWWKH WLPH 7S E\ ZKLFK WKH FXPXODWLYH DGRSWLRQ UDWH ZRXOG VXUSDVV D JLYHQ GHVLUDEOH WKUHVKROG YDOXH S JLYHQ WKH DGRSWLRQ KLVWRU\ )RUPDOO\ ILQG HVWLPDWH IRU ( 7S _ W W « WN ZKHUH 7S LV WKH WLPH E\ ZKLFK WKH FXPXODWLYHSHQHWUDWLRQUDWHKDVUHDFKHGWKHYDOXHSDQGDGRSWLRQKLVWRU\ LV JLYHQ E\ N WLPHSRLQWV RI WKH WLPHVHULHV ^ WS WS « WN SN ` )RUH[DPSOHIRUHFDVW7WKHWLPHE\ZKLFKKDOIWKHWDUJHWSRSXODWLRQ KDVDOUHDG\DGRSWHGWKHWHFKQRORJLFDOLQQRYDWLRQ
&+$37(5)25(&$67,1*66+$3('',))86,21352&(66(6
F(x), P(t)
f(x), dP(t)
0.8
0.12 0.1 0.08 0.06 0.04 0.02
0.6 0.4 0.2
2
4
6
8
10
x,t
2
4
6
8
10
x,t
)LJXUH$W\SLFDO66KDSHGFXUYH DVDSSOLHGWRDGLVWULEXWLRQIXQFWLRQDQGWRD GLIIXVLRQSURFHVV
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
3W I W εW
ZKHUH I W LVD IXQFWLRQ RI WLPH ZKLFK GHVFULEHV D VLJPRLG FXUYH DQG WKHHUURUWHUPεWLVFRQVLGHUHGWREHKRPRVFHGDVWLF KRPRJHQRXVWLPH LQGHSHQGHQWYDULDQFH QDPHO\( εW 9DU εW σε 7KH VHFRQG FDWHJRU\ FRPSULVHV OLQHDUL]HG WUHQG FXUYHV ZKLFK DVVXPHWKDWVRPHQRQOLQHDUWUDQVIRUPDWLRQRI3WPD\EHH[SUHVVHGDVD OLQHDUIXQFWLRQRIW
I 3W DEWεW
5(63216(02'(/,1*0(7+2'2/2*<
7KLV DSSURDFK PD\ EH OLNHQHG WR WKH UHVSRQVHQRUPDOL]LQJ DSSURDFK XVHG IRU OLQHDU UHJUHVVLRQ DQDO\VLV RU WR WKH OLQN IXQFWLRQ XVHG LQ JHQHUDOL]HGOLQHDUPRGHOV */0 1RWHKRZHYHUWKDWIRUWKHODWWHULWLVD QRQOLQHDU WUDQVIRUPDWLRQ RI WKH UHVSRQVH PHDQ WKDW LV DVVXPHG WR EH H[SUHVVLEOHLQWHUPVRIWKHOLQHDUSUHGLFWRU,QRXUFDVH DVLQWKH QRUPDOL]LQJ DSSURDFK WKH QRQOLQHDU WUDQVIRUPDWLRQ LV DSSOLHG WR WKH GDWD^3W` QRWWRWKHPHDQIXQFWLRQ 0,QRWH EDVHGRQ
3W I 3W εW
2FFDVLRQDOO\ PRGHOV RI WKH WKLUG JURXS DUH REWDLQHG IURP WKRVH RI WKHILUVWE\VHWWLQJ
G3W 3W3W ∂I W ∂WεW
RUIURPWKRVHRIWKHVHFRQGFDWHJRU\E\VHWWLQJ >GI 3W G3W@>G3WGW@ E
D
IURPZKLFKZLWKGW ZHREWDLQDSSUR[LPDWHO\
G3W 3W3W E>GI 3W G3W@εW .
E
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
&+$37(5)25(&$67,1*66+$3('',))86,21352&(66(6
*LYHQDGLVWULEXWLRQIXQFWLRQ) [ ZKHUH;LVWKHUDQGRPYDULDEOHDSORW RI ) [ YV [ ZRXOG PRVW OLNHO\ UHYHDO D W\SLFDO VLJPRLGVKDSHG FXUYH VLPLODUWRWKDWGLVSOD\HGLQ)LJXUH,IZHVXEVWLWXWH3WIRU) [ DQGW IRU [ UHODWLRQVKLSV WKDW KDYH RULJLQDOO\ EHHQ GHULYHG WR GHSLFW GLVWULEXWLRQIXQFWLRQVPD\VHUYHWRGHVFULEHWKHUHODWLRQVKLS3W I W $VD UHVXOW VRPH ZHOO NQRZQ DQG FRPPRQO\ DSSOLHG GLVWULEXWLRQV OLNH WKH QRUPDOWKHORJLVWLFWKHH[WHQGHGORJLVWLF*RPSHUW]WKHORJQRUPDODQG :HLEXOODUHURXWLQHO\XVHGWRPRGHO3WLQ6VKDSHGGLIIXVLRQSURFHVVHV 0,FODVVLI\WKHYDULRXVPRGHOVDFFRUGLQJWRWKHSRVLWLRQRIWKHSRLQW RILQIOH[LRQRIWKH6FXUYHVLQFHWKHWLPLQJRIWKLVSRLQWFRLQFLGHVZLWK WKH PD[LPXP UDWH RI GLIIXVLRQ RU SHQHWUDWLRQ 7KH\ GLVWLQJXLVK EHWZHHQ V\PPHWULF PRGHOV LQIOH[LRQ SRLQW DW 3W OLNH WKH QRUPDO DQG WKH ORJLVWLF PRGHOV QRQV\PPHWULF PRGHOV 3W EXW FRQVWDQW OLNH*RPSHUW]DQGOLQHDUL]HG*RPSHUW] DQGIOH[LEOHPRGHOV ZKHUHWKH LQIOH[LRQ SRLQW FDQ RFFXU DQ\ZKHUH ZLWKLQ WKH VDPH PRGHO LQFOXGLQJ 3W 7\SLFDOH[DPSOHVDUHWKHORJQRUPDO:HLEXOODQGWKHH[WHQGHG ORJLVWLF 0, GHQRWH WKHVH FODVVHV E\ FODVV ,, V\PPHWULF FODVV ,, DV\PPHWULF DQG&ODVV,, IOH[LEOH %DVHGRQDFDUHIXOO\VHOHFWHGVXEVHWRIVHYHQPRGHOVWKDWEHORQJWR WKH GLIIHUHQW FODVVHV RXW RI WKH RULJLQDO WZHQW\QLQH 0, LQWURGXFHWKH IROORZLQJFODVVLILFDWLRQ &ODVV,, V\PPHWULF 6LPSOH/RJLVWLF0DQVILHOG $5 &ODVV,, DV\PPHWULF *RPSHUW])OR\G $5 &ODVV,, IOH[LEOH &ODVV,,$ :HLEXOO([WHQGHG/RJLVWLF % &ODVV,,% &XPXODWLYH/RJQRUPDO &ODVV ΙΙ3 LV VXEGLYLGHG EDVHG RQ UHVXOWV IURP VWHSZLVH GLVFULPLQDQW DQDO\VLVZKLFKKDVVKRZQWKDWWKHORJQRUPDOEHKDYHVGLIIHUHQWO\IURP WKH RWKHU IOH[LEOH FXUYHV WKHUHLQ S 7KH PRGHOV ZKLFK PD\ VHUYHIRUFRPSDULVRQZLWKWKHQHZDSSURDFKDUHGHWDLOHGLQ$SSHQGL[$ ZKHUHWKHPRGHOVDUHQXPEHUHGIRUHDV\UHIHUHQFH $PDMRULPSOLFDWLRQRIWKLVFODVVLILFDWLRQLVWKDWGLIIHUHQWGDWDVHWV RU GLIIHUHQWDVVXPSWLRQVUHJDUGLQJWKHVHVHWV ZRXOGUHTXLUHWKHDGRSWLRQRI GLIIHUHQW PRGHOV WKDW ZRXOG UHIOHFW WKH DFWXDO FRQGXFW RU DVVXPSWLRQV
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5)25(&$67,1*66+$3('',))86,21352&(66(6
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Φ ]3 DQG ) DQG Φ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
5(63216(02'(/,1*0(7+2'2/2*<
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η=1 UHLQWURGXFHG KHUH IRU FRQYHQLHQFH < H[S^ αλ > ε1 λ@µε2`
H[S^ αλ > σε1= λ@µσε2[ρ= ρ =@`
: ORJ < αλ > σε1= λ@µσε2[ρ= ρ =@ ,QWURGXFLQJ ρ=±1, ZH PD\ REWDLQ IURP WKH RULJLQ ,17 IRU VLPSOLFLW\ WKH VXEVFULSW S XVHG HDUOLHU WR GHQRWH WKH SWK TXDQWLOH LV KHQFHIRUWKGHOHWHG 7KHRULJLQ,17LV
\ H[S^ αλ > σε1] λ@µσε2ρ]`
Z αλ > σε1] λ@µσερ]@
)URPWKHVHRIIVSULQJ,17VPD\EHGHULYHG UHIHUWR&KDSWHU \ 0 H[S^ Β >H[S &] @ ']`
\ 0 H[S^ Β >H[S &] @`
\ 0 H[S^ Β& > $] &@`]!$
\ 0 H[S>$%] $] @
\ 0>] %& @%
ZKHUH0LVWKHUHVSRQVHPHGLDQ ]
&+$37(5)25(&$67,1*66+$3('',))86,21352&(66(6
1RWH WKDW LV GHULYHG IURP E\ DVVXPLQJ LQ WKH ODWWHU & 7KHSDUDPHWHU$LVWKHQUHSODFHGE\ %& VRWKDWWKHLQYHUVHRI UHVXOWVLQD%R[&R[WUDQVIRUPDWLRQ UHIHUWR6HFWLRQZKHUH WKLVWUDQVIRUPDWLRQLVDGGUHVVHG $OVRQRWHWKDWIRU ]FDQ EH H[SUHVVHG H[SOLFLWO\ LQ WHUPV RI \ 7KLV LPSOLHV WKDW WKH LQYHUVH RI WKHVHH[SUHVVLRQVPD\EHXVHGDVQRUPDOL]LQJWUDQVIRUPDWLRQV 17V %RWK ,17V DQG 17V ZLOO ODWHU EH HPSOR\HG IRU IRUHFDVWLQJ 6HFWLRQ 7KHIROORZLQJ17VZLOOEHXVHG 0RGHO17>IURP @ ] $ ORJ \0 >%ORJ \0 @ 0RGHO17>IURP @ ] $% > \0 %@
0RGHO 17 LV D %R[&R[ QRUPDOL]LQJ WUDQVIRUPDWLRQ ZLWK D UH VFDOHG\VRWKDWWKHPHGLDQVRI<DQG=DUHSUHVHUYHGQDPHO\IRU\ 0 WKHPHGLDQ ZHKDYH] 2WKHU17VPD\EHVLPLODUO\GHULYHGIURPWKH ,17VJLYHQLQ ,WLVQRZSRVVLEOHWRGHILQHIRUPDOO\WKHSURFHGXUHVWRILWDPRGHODQG XVH LW IRU IRUHFDVWLQJ 7KH SURFHGXUHV DUH EDVHG RQ WKH FRUUHVSRQGHQFH EHWZHHQ^]\`WKHTXDQWLOHVLQWKHTXDQWLOHUHODWLRQVKLSVJLYHQDERYHE\ WKH ,17V > @ DQG E\ WKH 17V > @ DQG ^W 3W` ZKHUH 3W LV WKH FXPXODWLYH UHODWLYH GHPDQG DW WLPH W 7R ILW DQ\ RI WKH ,17V RU17V WRDWLPHVHULHVUHSUHVHQWLQJFXPXODWLYHUHODWLYHGHPDQG ZH WUDQVIRUP WKH JLYHQ GDWD E\ WZR GLIIHUHQW SURFHGXUHV HDFK UHSUHVHQWLQJDGLIIHUHQWIRUHFDVWLQJREMHFWLYH 3URFHGXUH, )RUD*LYHQ7LPH6HULHV^3W`0RGHODQG)RUHFDVW73LQ 7HUPVRIDVSHFLILHG3 7KLVUHTXLUHVILWWLQJDQ,17WRWKHJLYHQGDWD7KHIROORZLQJSURFHGXUHLV SXUVXHG
5(63216(02'(/,1*0(7+2'2/2*<
$ )LWWLQJWKHPRGHO
5HSODFH 3 DVVRFLDWHG ZLWK WKH ,17 ZLWK SW DQG ILQG WKH FRUUHVSRQGLQJ]W7KLVLPSOLHVILQGLQJIRUHDFKSW
Φ SW ]W ZKHUHΦ LVWKHVWDQGDUGQRUPDOLQYHUVH&') 5HSODFH\LQWKHH[SUHVVLRQIRUWKH,17E\W )LWDQ,17WRWKHQHZGDWDVHW^]WW`XVLQJQRQOLQHDUOHDVWVTXDUHV 1//6
% )RUHFDVW73JLYHQ3
)LQGΦ 3 ] ,QWURGXFH]LQWRWKHILWWHG HVWLPDWHG ,17WRIRUHFDVW 73
6XPXS )LWWLQJDQ,17WRDJLYHQWLPHVHULHVDOORZVXVWRIRUHFDVWWKH WLPH 73 ZKHQ FXPXODWLYH GHPDQG LV H[SHFWHG WR FURVV D GHVLUDEOH WKUHVKROGYDOXH3 3URFHGXUH,, )RUD*LYHQ7LPH6HULHV^3W`0RGHODQG)RUHFDVW37 LQ7HUPVRIDVSHFLILHG7 7RIRUHFDVW37JLYHQ7DQRUPDOL]LQJWUDQVIRUPDWLRQ 17 QHHGVWREH HVWLPDWHGQDPHO\DTXDQWLOHUHODWLRQVKLSWKDWH[SUHVVHV]LQWHUPVRI\ 7KH DQDORJ\ EHWZHHQ WKH QRUPDOL]LQJ WUDQVIRUPDWLRQ ] I \ DQG WKH HVWLPDWLRQDQGIRUHFDVWLQJSUREOHPLVDVIROORZV $ )LWWLQJWKHPRGHO
)RUHDFKREVHUYDWLRQWUHSODFH\LQWKH17ZLWKW )RUHDFKREVHUYDWLRQSWUHSODFH]LQWKH17ZLWK]W Φ−1 SW )LWDQ17WRWKHQHZGDWDVHW^W]W`XVLQJ1//6
&+$37(5)25(&$67,1*66+$3('',))86,21352&(66(6
% )RUHFDVW37JLYHQ7
5HSODFH\LQWKH17ZLWK7 8VHWKHILWWHG17WRILQG]7 I 7 )LQG3W Φ ]7
7KHFRUUHVSRQGHQFHEHWZHHQWKHRULJLQDOYDULDEOHVRIWKH,17DQGWKH 17 DQG WKH WLPH VHULHV YDULDEOHV ^W SW` LV JLYHQ E\ WKH IROORZLQJ IXQGDPHQWDOWUDQVIRUPDWLRQVZKLFKVXPPDUL]HWKHIRUHFDVWLQJSURFHVV
*LYHQ3IRUHFDVW7 ZLWKDILWWHG,17 3 Φ ]7 Æ]7Æ7
*LYHQ7IRUHFDVW3 ZLWKDILWWHG17 7Æ]7ÆΦ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
5(63216(02'(/,1*0(7+2'2/2*< 7DEOH6RXUFH'DWDDQG)RUHFDVWVEDVHGRQ,17V WRIRUHFDVW73 DQG17V WRIRUHFDVW37 )RUHFDVWLQJ37 *LYHQ7
3HULRG &XPXODWLYHGHPDQG )RUHFDVWLQJ73 *LYHQ3 W 3URSU 3W
]WYDOXHV ]WYDOXHV H[DFW IURPILWWHG 17
66KDSHG3ORWRI'DWD6HW
6HULHV
&XPXODWLYH'HPDQG SURSRUWLRQ
3HULRG 7
)LJXUH3XUFKDVHRI% :79VHWVLQWKH86$
&+$37(5)25(&$67,1*66+$3('',))86,21352&(66(6
ZKHUH
W 0 H[S^ Β >H[S &] @ ']`
0 % & '
5HVXOWV IURP WKH ILWWHG ,17 DUH JLYHQ LQ 7DEOH 'HYLDWLRQV RI IRUHFDVWV RI 7 DUH UHVWULFWHG WR DERXW ±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
W 0 H[S Β] '] 0 H[S β]
ZKHUH0DQGβDUHWZRSDUDPHWHUVWKDWQHHGWREHHVWLPDWHG:HUHDOL]H WKDW DQDO\]LQJ WKH GDWD YLD WKH ,17 SRLQWV WR WKH ORJQRUPDO PRGHO DV SUREDEO\WKHEHVWILWWLQJPRGHO7KHUHDGHUPD\ZLVKWRUHILWWKLVPRGHO 5HVXOWVVKRZQLQ %HQVRQDQG 6KRUH FOHDUO\ GHPRQVWUDWH YLD VHYHUDO HIIHFWLYHQHVV PHDVXUHV DSSOLHG WR YDULRXVO\VL]HG GDWDVHWV WKDW WKH,17VPD\VHUYHDVSRZHUIXOIRUHFDVWLQJWRROVIRU6VKDSHGGLIIXVLRQ SURFHVVHV
5(63216(02'(/,1*0(7+2'2/2*<
)RUHFDVWLQJ37JLYHQ7 5HODWLQJWRWKHVDPH% :79GDWDLWLVUHTXLUHGWRSUHSDUHIRUHFDVWVIRU 37WKHFXPXODWLYHGHPDQGSURSRUWLRQH[SHFWHGDWWLPHSHULRG7)LWWLQJ 17 ZHREWDLQ
] $% > W0 %@
ZLWKHVWLPDWHGSDUDPHWHUV
0 $ %
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
&+$37(5)25(&$67,1*66+$3('',))86,21352&(66(6
WUDQVIRUPDWLRQ,Q$GYDQFHVLQ0HWKRGRORJLFDODQG$SSOLHG$VSHFWVRI3UREDELOLW\ DQG 6WDWLVWLFV %RRN 1 %DODNULVKQDQ (G 9 SS *RUGRQ DQG %UHDFK6FLHQFH3XEOLVKHUV&DQDGD > @
$SSHQGL[$&XUUHQW)RUHFDVWLQJ0RGHOV 6LPSOH/RJLVWLF 3W
$ F W E f F f $ F m H[S EW
*RPSHUW] 3W H[S F m H[S EW $ F W E f F f &XPXODWLYH/RJQRUPDO W
3W T
OQ \ H[S G\ $ FW ! f ! \ n !
W :HLEXOO 3W H[S $ FW o f f
o
([WHQGHG/RJLVWLF %
S H[S S $ T W T 3W $ F W S f T f $ F H[S S $ T W F
0DQVILHOG $5 3W 3W E3W 3W $ F W E f )OR\G $5 3W 3W E3W 3W $ F W E f
5(63216(02'(/,1*0(7+2'2/2*<
$SSHQGL[%'HVFULSWLRQRI'DWD6HWV UHIHUIRUUHOHYDQWUHIHUHQFHV WR0, VHW
3URGXFWRULQQRYDWLRQ &RORXU7986$ &$7986$ 79% :86$ 79% :'HQPDUN 79% :)UDQFH 79% :)LQODQG 79% :1RUZD\ 79% :6ZHGHQ 3URGXFW 3URGXFW 3URGXFW 3URGXFW 3URGXFW 3URGXFW &HQWUDO+HDWLQJ8. 5DQJHV86$ &RIIHHPDNHU86$ +HDWLQJSDGV86$ +RWSODWHV86$ :DWHUKHDWHUV86$ %HGFRYHULQJV86$ %OHQGHU86$ 'LVSRVHU86$ )UHH]HU86$ 0L[HU86$ 5DQJHVEXLOWLQ86
VL]H
2ULJLQDOVRXUFH /HHHWDO /HHHWDO /LQGEHUJ /LQGEHUJ /LQGEHUJ /LQGEHUJ /LQGEHUJ /LQGEHUJ 5DOSKDQG/LQIRUG 5DOSKDQG/LQIRUG 5DOSKDQG/LQIRUG 5DOSKDQG/LQIRUG 5DOSKDQG/LQIRUG 5DOSKDQG/LQIRUG 0RQWKO\'LJHVWRI6WDWLVWLFV8. 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\ 0HUFKDQGLVLQJ:HHN)HEUXDU\
&+$37(5)25(&$67,1*66+$3('',))86,21352&(66(6
$SSHQGL[% &RQW G
VHW
3URGXFWRULQQRYDWLRQ (OHFWURQLFVZLWFKLQJV\VWHP8. (OHFWURQLFVZLWFKLQJV\VWHP86$ 0DQPDGHILEUH86 7HOH[6LQJDSRUH &RPSXWHU6LQJDSRUH 6WHDP9HVVHO86$ &DW&UDFNLQJ RLO
VL]H
2ULJLQDOVRXUFH %ULWLVK7HOHFRP /HHHWDO 0DUWLQR 4XDGGXV 4XDGGXV 5DOSKDQG/LQIRUG /LQVWRQHDQG6DKDO
6WHHO2[\SURFHVV IRU6SDLQ 6WHHO2[\SURFHVV IRU:RUOG 6WHHO2[\SURFHVV IRU%HOJLXP 6WHHO2[\SURFHVV IRU)UDQFH 6WHHO2[\SURFHVV IRU1HWKHUODQGV 6WHHO2[\SURFHVV IRU6ZHGHQ 3ODVWLFJDOPLON 3ODVWLFJDOPLON 0DQPDGHILEUH8. 86WHOHSKRQHV 'RPHVWLF(OHF86 'LVKZDVKHU86 &ORWKHVGU\HU86 0HWDO9HVVHOV86
3R]QDQVNL 3R]QDQVNL 3R]QDQVNL 3R]QDQVNL 3R]QDQVNL 3R]QDQVNL
This page intentionally left blank
3DUW,9
02'(/,1*5$1'209$5,$7,21 $33/,&$7,216
This page intentionally left blank
&KDSWHU
500'LVWULEXWLRQDO$SSUR[LPDWLRQV
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
318
RESPONSE MODELING METHODOLOGY
19.2. Fitting RMM with Normal or Log-normal Errors In this section the RMM quantile function is fitted to a set of known statistical distributions, pursuing the original formulation of the RMM model, namely, the error terms, ε1 and ε2, are either normal or lognormal. The RMM error quantile-function is fitted using the "QuantileMatching" procedure (refer to Section 10.3 for details). We opt for this approach because when the distribution modeled by RMM is completely specified (no sample estimates are used in the fitting procedure), quantile-matching seems to be most convenient for identifying numerically the RMM parameters. The resulting goodness-of-fit is judged (in this chapter) mainly by how well the first four moments of the fitted RMM error distribution match the exact moments. The moments of the fitted quantile function are numerically evaluated using the simple expression for the r-th non-central moment µr' =
∫
∞
−∞
y ( z ) r φ ( z )dz ,
(19.1)
where y(z) is the fitted quantile function, and φ is the d.f. of the standard normal distribution. From (19.1), the mean, the variance, the skewness measure, Sk, and the kurtosis measure, Ku, are (refer to Stuart and Ord, 1987, Section 3.14) µ = µ1', V(Y) = µ2' - (µ1' )2 , Sk = ¥β1 = E[(Y-µ)3] / σ3 = [ µ3' - 3 µ2' µ1' + 2 (µ1')3] / σ3 , Ku = β2 - 3 = E[(Y-µ)4] / σ4-3 = [µ4' - 4 µ3' µ1' + 6 µ2' (µ1')2 - 3 (µ1')4] / σ4 - 3 .
(19.2)
For the normal distribution: Sk=0, Ku=0. Two versions of the quantile-function are employed in the fitting procedure (refer to Chapter 9): Y = (M) exp{(α/λ)[(1+ε1) λ - 1] + ε2 }, (19.3) Y = (M) exp{(α/λ)[ exp(λε1 ) - 1] + ε2},
(19.4)
&+$37(5 500',675,%87,21$/$3352;,0$7,216
ZKHUH ^0 α, λ` DUH SDUDPHWHUV WKH HUURU WHUPV ε1 DQG ε2, DUH IURP D ELYDULDWH QRUPDO GLVWULEXWLRQ ZLWK VWDQGDUG GHYLDWLRQV σε1 DQG σε2, UHVSHFWLYHO\DQGFRUUHODWLRQρ, WKDWLV ε1 σε1=ε2 σε2 [ρ= ρ =@
ZKHUH=DQG=DUHLQGHSHQGHQWVWDQGDUGQRUPDOYDULDEOHV 7KH500TXDQWLOHIXQFWLRQV DQG KDYHWZRHUURUWHUPV 7RILWDXQLYDULDWHGLVWULEXWLRQLQWURGXFHρ=±LQWR %RWKTXDQWLOH IXQFWLRQV DUH QRZ H[SUHVVHG LQ WHUPV RI D VLQJOH VWDQGDUG QRUPDO YDULDEOH7KLVSURSHUW\LVKHOSIXOLQILWWLQJWKHTXDQWLOHIXQFWLRQVYLDWKH TXDQWLOHPDWFKLQJSURFHGXUH 5HVFDOLQJ WKH SDUDPHWHU λ LQ WHUPV RI σε1, QDPHO\ λ λσε1 UHVFDOLQJ HQVXUHV UHDVRQDEO\ VPDOO YDOXHV IRU λ0) ZH REWDLQ IURP IRUWKHTXDQWLOH\LQWHUPVRIWKHFRUUHVSRQGLQJVWDQGDUG QRUPDOTXDQWLOH]
\ 0 H[S^ σε1αλ0 > σε1] (λσε1)@ (σε2ρ)]`
\ 0 H[S^ σε1αλ0 >H[S λ] @ (σε2ρ)]`
ZKHUH0LVWKHPHGLDQ ] DQGρ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` ZLWK FRUUHVSRQGLQJ TXDQWLOH YDOXHV RI WKH DSSUR[LPDWHGGLVWULEXWLRQDUHLGHQWLILHG7KHVHDUHWKHQLQFRUSRUDWHGLQ D QRQOLQHDU OHDVWVTXDUHV 1//6 SURFHGXUH WR GHWHUPLQH WKH SDUDPHWHUVRIWKHILWWHGTXDQWLOHIXQFWLRQ*RRGQHVVRIILWLVDVVHVVHGE\ FDOFXODWLQJ QXPHULFDOO\ IURP WKH ILWWHG TXDQWLOH IXQFWLRQ WKH ILUVW IRXU PRPHQWVQDPHO\WKHPHDQWKHYDULDQFHDQGWKHVNHZQHVVDQGNXUWRVLV
5(63216(02'(/,1*0(7+2'2/2*<
PHDVXUHV>XVLQJ DQG @1XPHULFDOLQWHJUDWLRQH[WHQGVIURP 3 ] WR3 ] :HQRWHKHUHWKDWWKHUHDUHRWKHU'LVWDQFHPHDVXUHVLQWKHOLWHUDWXUH WKDW DLP WR MXGJH KRZ ZHOO RQH GLVWULEXWLRQ ID \ PD\ PRGHO DSSUR[LPDWH DQRWKHU JLYHQ GLVWULEXWLRQ I 2IWHQ WKLV GLVWDQFH LV HYDOXDWHG YLD QRQQHJDWLYH GLYHUJHQFH RU SVHXGRGLVWDQFH PHDVXUHV GHVLJQDWHG E\ ' ID I )RU H[DPSOH .DULDQ DQG 'XGHZLF] XVH WKH/SQRUP
__ID \ I \ __S ∫_ID \ I \ _SG\ SS≥
ZLWKS RUWRDVVHVVWKHTXDOLW\RIWKHILWRIWKHJHQHUDOL]HG/DPEGD GLVWULEXWLRQ2WKHUFRPPRQO\XVHGGLVWDQFHPHDVXUHVDQGDJRRGUHYLHZ RIFXUUHQWOLWHUDWXUHPD\EHIRXQGLQ:DULFN $SSO\LQJ 1//6 WR WKH TXDQWLOH GDWDVHWV JHQHUDWHG IURP WKH VDPSOH RI GLVWULEXWLRQV GHVFULEHG HDUOLHU SDUDPHWHUV DQG FDOFXODWHG PRPHQWV DUH REWDLQHG 7KHVH DUH GLVSOD\HG LQ 7DEOH 7KH SDUDPHWHUV VKRZQ UHODWH WR WKH IROORZLQJ UHSDUDPHWHUL]HG TXDQWLOH IXQFWLRQV
\ 0 H[S^>Β/(&$ @> $] (&$ @ ']`
\ 0 H[S^%>([S &] @']`
D D
ZKHUH0LVWKHH[DFWPHGLDQ>2QWKHUHDVRQIRUH[SUHVVLQJWKHH[SRQHQW E\ &$ UHODWHWRWKHFRPPHQWSUHFHGLQJ @ ,Q 7DEOH H[DFW PRPHQWV DUH LQ WKH XSSHU HQWULHV LQ EROG 6HFRQGDQGWKLUGHQWULHVIURPWKHWRSUHODWHWRILWWLQJE\ D DQGE\ D UHVSHFWLYHO\ ZLWK = QRUPDO /RZHU HQWULHV UHODWH WR ILWWLQJ D ZLWKDORJLVWLFHUURUWHUP WKLVZLOOEHGLVFXVVHGLQ6HFWLRQ ([DPLQLQJ 7DEOH RQH UHDOL]HV WKDW DOO ILUVW IRXU PRPHQWV DUH ZHOOSUHVHUYHGLQWKHILWWHGTXDQWLOHIXQFWLRQV1RWHWKDWRQO\IRUWKHEHWD GLVWULEXWLRQWKHSDUDPHWHU'LVQHJDWLYHLPSO\LQJ ρ )RUWKH)UDWLR GLVWULEXWLRQ WKH YDOXH RI $ LV RI WKH RUGHU RI LPSO\LQJ WKDW WKH ILWWHG500LQIDFWDSSUR[LPDWHVWKH)UDWLRGLVWULEXWLRQE\DORJQRUPDO GLVWULEXWLRQ )RU WKH QRUPDO D SURYLGHV DQ H[DFW H[SUHVVLRQ ZKHUHIRUH WKH SDUDPHWHUV LQ 7DEOH DUH WKH H[DFW RQHV $
&+$37(5 500',675,%87,21$/$3352;,0$7,216
7DEOH 3DUDPHWHUVDQGPRPHQWV H[DQGDSS IRUWKHILWWHGTXDQWLOHIXQFWLRQV ([DFWPRPHQWVDUHLQEROG 'LVWULEXWLRQ
$
1RUPDO µ σ %HWD α β 0 )5DWLR Q Q 0 *DPPD α β 0 *DPPD α β 0
*DPPD α β 0 :HLEXOO α β 0 :HLEXOO α β H[SGLV 0 :HLEXOO α β 0 :HLEXOO α β 0
3DUDPHWHUV % &
'
0HDQ
0RPHQWV 9DU 6N .X
5(63216(02'(/,1*0(7+2'2/2*< 7DEOH FRQW G
'LVWULEXWLRQ $ % & 5D\OHLJK σ 0 ([WUHPH9DOXH α β 0
'
0HDQ
9DU
6N
.X
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
] 8 ORJ> 3 3 @
&+$37(5 500',675,%87,21$/$3352;,0$7,216
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
5(63216(02'(/,1*0(7+2'2/2*<
LQWXLWLYHO\ HYLGHQW $SSUR[LPDWLQJ WKH &') RI WKH QRUPDO GLVWULEXWLRQ DQGWKHTXDQWLOHVRIDGLVFUHWHUYDUHWZRVXFKFDVHV ,Q 6HFWLRQ WKH DSSUR[LPDWLRQ IRU WKH 3RLVVRQ TXDQWLOH LV GHYHORSHGDQGLQ6HFWLRQ WKHDSSUR[LPDWLRQVIRUWKH&')RIWKH VWDQGDUGQRUPDOGLVWULEXWLRQ $SSUR[LPDWLQJWKH3RLVVRQTXDQWLOH )LWWLQJ D YLD1//6ZLWKVDPSOHVRI3RLVVRQTXDQWLOHVDQG3RLVVRQ SDUDPHWHUλ ^`SDUDPHWHUV YDOXHVDQGDSSUR[LPDWH TXDQWLOHYDOXHVDUHREWDLQHGDVVKRZQLQ7DEOH 7RFODULI\KRZWKHH[DFWTXDQWLOHYDOXHVZHUHSUHSDUHGIRUWKHILWWLQJ SURFHGXUH VXSSRVH WKDW λ )RU WKH 3RLVVRQ TXDQWLOH YDOXH \ ZH KDYH3 7KHFRUUHVSRQGLQJVWDQGDUGQRUPDOTXDQWLOHLV] 7KH WZR TXDQWLOHV ^` FRPSULVH RQH SRLQW LQ WKH VDPSOHVHW XVHGLQWKH1//6SURFHGXUH 2EVHUYLQJ7DEOH ZHQRWHWKHKLJKDFFXUDF\IRUVPDOOλDQGWKH KLJK DFFXUDF\ WKDW LV SUHVHUYHG IRU H[WUHPH WDLO YDOXHV OLNH \ IRU λ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λ
λ
3$5$0(7(56 0 % &
'
48$17,/(6
&+$37(5 500',675,%87,21$/$3352;,0$7,216
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i ] D V $ D V $ D V ] p Φ ] i ] D V $ D V $ D V ] f ZKHUHΦ ] LVWKHVWDQGDUGQRUPDO&')DQG V S_]_ S D
D D
7KLVDSSUR[LPDWLRQKDVDQDEVROXWHHUURUOHVVWKDQ 0RUHUHFHQWO\%DJE\ VXJJHVWHGWKHIROORZLQJDSSUR[LPDWLRQ
Φ ] ± ^ >H[S ] H[S ] √
(π ] H[S ] @`
ZKHUHDPLQXVVLJQUHIHUVWR]7KHDEVROXWHHUURULVOHVVWKDQ 500EDVHG DSSUR[LPDWLRQV DUH QRZ GHYHORSHG ,Q 6KRUH D HTXDWLRQ D ZDVXVHGWRGHULYHDQDSSUR[LPDWLRQWRWKH&')RIWKH VWDQGDUG QRUPDO GLVWULEXWLRQ 7KH SRLQW RI GHSDUWXUH ZDV DQ 500 PRGHOLQJ RI WKH H[SRQHQWLDO TXDQWLOH IXQFWLRQ ,QWURGXFLQJ IRU WKH TXDQWLOHRIWKHH[SRQHQWLDOGLVWULEXWLRQ ZLWKSDUDPHWHUλ LQWHUPVRI WKHFRUUHVSRQGLQJ&')) \ 3 H[S \ LQWR D ZHREWDLQ
5(63216(02'(/,1*0(7+2'2/2*<
\ ORJ 3 0 H[S^%>H[S &] ']`
)URPWKLVDQH[SOLFLWH[SUHVVLRQIRU3LQWHUPVRIWKHFRUUHVSRQGLQJ VWDQGDUGQRUPDOTXDQWLOH]LV 3 ] ±H[S^ORJ H[S>%>H[S &] @']@` ZKHUHZHLQWURGXFHGµ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Φ ] ≅ ] ] ] ] ] ] $ $ $ 8 8 8 8 8 8 $
] 8
] 8
$
] 8
$ 2>]@
%\FRQWUDVWD7D\ORUH[SDQVLRQRI ZLWKSDUDPHWHUVJLYHQLQ UHVXOWVLQWKHIROORZLQJH[SDQVLRQ 3 ] ≅ ]] ] ]
]]]]
] ]( ]2>]@
&+$37(5 500',675,%87,21$/$3352;,0$7,216
)LJXUH (UURUSORW DSSPLQXVH[ RIDSSUR[LPDWLRQ ZLWKSDUDPHWHUVJLYHQ E\ ]!
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η LQ GHULYLQJ IURP WKH RULJLQDO500PRGHO ZHREWDLQ J ] H[S^ORJ H[S>%>H[S $&] @']@` ZLWKWKHILWWHGSDUDPHWHUV YDOXHV
5(63216(02'(/,1*0(7+2'2/2*<
$ %
& '
,QWURGXFLQJ IURP LQWR DQG WKHQFH LQWR ZH REWDLQDQDSSUR[LPDWLRQIRUWKH&')RIWKHVWDQGDUGQRUPDOGLVWULEXWLRQ WKHHUURUSORW RI ZKLFKLVJLYHQ LQ )LJXUH 7KHPD[LPXP DEVROXWH HUURUUHGXFHGIURP UHIHUWR)LJXUH WR )XUWKHUPRUH D 7D\ORU H[SDQVLRQ RI WKH DSSUR[LPDWLRQ VKRZV WKDW WKH FRHIILFLHQWV RI DUHZHOOSUHVHUYHGLQWKHH[SDQVLRQRI
)LJXUH (UURUSORWRIDSSUR[LPDWLRQ ZLWKSDUDPHWHUVJLYHQE\ ,Q D VLPLODU IDVKLRQ DQ 500EDVHG DSSUR[LPDWLRQ IRU WKH VWDQGDUG QRUPDO&')PD\EHGHULYHGIURP D 5HSDUDPHWHUL]HGZHKDYHIRU J ] ZLWKSDUDPHWHUV^α, λ, 66` J ] H[S^ORJ H[S^>α λ6)][(1+6] λ/6)@6]` ([SDQGLQJ LQWRD7D\ORUVHULHVZHUHDOL]HWKDWWRSUHVHUYHWKH FRHIILFLHQW RI ] LQ WKH H[SDQVLRQ RI WKH H[DFW QRUPDO &') ZH QHHGWRKDYH
α > π ORJ ±6] / 6
,QWURGXFLQJ EDFN LQWR DQG WKHQFH LQWR DQ DSSUR[LPDWLRQ ZLWK WKUHH SDUDPHWHUV ^λ, 6 6` LV REWDLQHG )LWWLQJ VLPLODUO\WRWKHHDUOLHUILWWLQJWKHSDUDPHWHUV YDOXHVDUH
CHAPTER 19 RMM DISTRIBUTIONAL APPROXIMATIONS
329
λ = -0.61228883, S1 = -0.11105481, S2 = 0.44334159, α = -6.37309208.
(19.22)
[the value of α was derived from (19.21) with the above values of S1 and S2; Also the coefficients of z in (19.20), S1 and S2, have different signs, a possibility that is anticipated from (19.7)]. Introducing from (19.22) into (19.20) and thence into (19.17), the error plot in Figure 19.4 is obtained. The maximum absolute error is reduced from 2(10)-6 (Figure 19.3) to 6(10)-7. Although (19.20) has only three parameters that need to be determined from the fitting procedure (α is not), it has accuracy better than the four- parameter approximation (19.18). As a final comment, we note two interesting features of the new approximations for the normal CDF. First, they are non-polynomial. This implies that their high accuracy emanates not from the large number of parameters that characterize polynomial approximations (refer to Johnson et al., 1994), but possibly because the RMM original model is in some way related to the normal distribution. This shows in that the accuracy obtained from
6 × 10-7 4 × 10-7 2 × 10-7 -7.5
-5
-2.5 -2 × 10-7
2.5
5
7.5
-4 ×10-7 -6 × 10-7 Figure 19.4. Error plot (app. minus ex.) of approximation (19.20) with parameters given by (19.22) (-9
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5 500',675,%87,21$/$3352;,0$7,216
^)5DWLR'LVWULEXWLRQ>@`
^%HWD'LVWULEXWLRQ>@`
^([WUHPH9DOXH'LVWULEXWLRQ>@` )LJXUH ([DFW DQG DSSUR[LPDWH ILWWHG TXDQWLOH IXQFWLRQV YV TXDQWLOH VHTXHQWLDO QXPEHU OHIW DQGWKHHUURUSORW YV]ULJKW IRUVRPHRIWKHGLVWULEXWLRQVLQ7DEOH &RQW GRQWKHQH[WWZRSDJHV
5(63216(02'(/,1*0(7+2'2/2*<
^&KL6TXDUH 'LVWULEXWLRQ>@`
^&KL6TXDUH'LVWULEXWLRQ>@` T
^:HLEXOO'LVWULEXWLRQ>@`
&+$37(5 500',675,%87,21$/$3352;,0$7,216
^:HLEXOO'LVWULEXWLRQ>@`
^:HLOEXOO'LVWULEXWLRQ>@`
^5D\OHLJK'LVWULEXWLRQ>@`
5(63216(02'(/,1*0(7+2'2/2*<
5HIHUHQFHV >@ $EUDPRZLW] 0 6WHJXQ , $ +DQGERRN RI 0DWKHPDWLFDO )XQFWLRQV 1DWLRQDO%XUHDXRI6WDQGDUGV1HZ@ %DJE\ 5 - &DOFXODWLQJ QRUPDO SUREDELOLWLHV $PHULFDQ 0DWKHPDWLFDO 0RQWKO\ >@ &DR - + &KHQ . ,QWURGXFWLRQ WR 5HOLDELOLW\ 0DWKHPDWLFV %HLMLQJ $FDGHPLF3XEOLVKHUV >@ -RKQVRQ 1 / .RW] 6 %DODNULVKQDQ 1 &RQWLQXRXV 8QLYDULDWH 'LVWULEXWLRQV²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¶V $GYDQFHG 7KHRU\ RI 6WDWLVWLFV 9 'LVWULEXWLRQ7KHRU\&KDUOHV*ULIILQ &RPSDQ\/WG/RQGRQ > @:DULFN- $GDWDEDVHGPHWKRGIRUVHOHFWLQJWXQLQJSDUDPHWHUVLQPLQLPXP GLVWDQFHHVWLPDWRUV&RPSXWDWLRQDO6WDWLVWLFVDQG'DWD$QDO\VLV,QSUHVV
&KDSWHU
,QYHUVH1RUPDOL]LQJ7UDQVIRUPDWLRQV
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
5(63216(02'(/,1*0(7+2'2/2*<
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αλ > ε1 λ@ε2`
ZKHUH^0α, λ`DUHUHDOYDOXHGSDUDPHWHUVDQGWKHHUURUWHUPVε1DQG ε2,DUHIURPDELYDULDWHQRUPDOGLVWULEXWLRQZLWKVWDQGDUGGHYLDWLRQVσε1 DQGσε2, UHVSHFWLYHO\DQGFRUUHODWLRQρ, QDPHO\ ε1 σε1=ε2 σε2 [ρ= ρ =@
ZKHUH=DQG=DUHLQGHSHQGHQWVWDQGDUGQRUPDOYDULDEOHV,QWURGXFLQJ IURP LQWR DQG DVVXPLQJ ρ ± EHFRPHV ZLWK = UHSODFHGE\=
< 0 H[S^ αλ > σε1= λ@ σε2ρ=`
(TXDWLRQ LV QRZ D IXQFWLRQ RI D VLQJOH VWDQGDUG QRUPDO YDULDEOHDQGWKHUHIRUHFDQEHSHUFHLYHGDVDQ,17:HGHQRWH WKH 2ULJLQ ,17 WR GLVWLQJXLVK LW IURP 2IIVSULQJ SDUDPHWHUUHGXFHG ,17V WR EH GHYHORSHG ODWHU ,Q 6HFWLRQ WKH FDSDELOLW\ RI WR UHSUHVHQW ZHOO D YDULHW\ RI GLIIHUHQWO\VKDSHG GLVWULEXWLRQV ZDV GHPRQVWUDWHG
&+$37(5,19(56(1250$/,=,1*75$16)250$7,216
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
5(63216(02'(/,1*0(7+2'2/2*<
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¶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
&+$37(5,19(56(1250$/,=,1*75$16)250$7,216
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
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5,19(56(1250$/,=,1*75$16)250$7,216
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
5(63216(02'(/,1*0(7+2'2/2*<
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α,λ,σε1σε2ρ` 2QH YDULDWLRQ RI WKH RULJLQ ,17 DOUHDG\ LQWURGXFHG LQ SUHYLRXV FKDSWHUV LV ZLWKρ= ±1) < 0 H[S^ αλ >H[S λε1 @ε2`
0 H[S^ αλ >H[S λσε1= @ σε2ρ=`
7KLV ,17 WRJHWKHU ZLWK LWV FRXQWHUSDUW EDVHG RQ FRQGLWLRQLQJ =_= ] UHIHU WR 6HFWLRQ PD\ EH UHZULWWHQ DV WKH TXDQWLOH IXQFWLRQ>UHIHUWR @ \ 0 H[S^$>H[S %] @']` ZKHUHHLWKHU%RU' EXWQRWERWK FDQEHQHJDWLYH,17 KDVRQO\ IRXUSDUDPHWHUVLQFOXGLQJWKHUHVSRQVHPHGLDQ0 WKHUHVSRQVHTXDQWLOH WKDWFRUUHVSRQGVWR] 7KHXQLTXHSURSHUWLHVRI ZHUHGLVFXVVHG LQ 6HFWLRQ DQG LWV FDSDELOLW\ WR UHSUHVHQW ZHOO GLYHUVHO\VKDSHG GLVWULEXWLRQV ZDV GHPRQVWUDWHG LQ 7DEOH 0RPHQWPDWFKLQJ SURFHGXUHVEDVHGRQORZGHJUHHPRPHQWVZHUHGHYHORSHGIRU,17,LQ 6XEVHFWLRQVDQG
&+$37(5,19(56(1250$/,=,1*75$16)250$7,216
2IIVSULQJ,17,, )URP 7DEOH LW LV FOHDU WKDW WKH SDUDPHWHU ' LQ ZKLFK UHSUHVHQWV σε2ρ) LV FRQVLVWHQWO\ VPDOO UDQJLQJ IURP IRU QHDU V\PPHWULF GLVWULEXWLRQV WR DURXQG IRU H[WUHPHO\ VNHZHG GLVWULEXWLRQV $VVXPLQJ ' HTXLYDOHQWO\ DVVXPLQJ WKDW σε2 LV QHJOLJLEOH)ZHREWDLQ,17,,
\ 0 H[S^ Α >H[S %] @`
:LWKDVSHFLILHGPHGLDQ 0 ,17,,KDVWZRSDUDPHWHUVWKDWQHHGWREH GHWHUPLQHG)LWWLQJ WRDVDPSOHVHWRIGLVWULEXWLRQVXVLQJWKHVDPH 1//6 TXDQWLOHEDVHG SURFHGXUH XVHG WR ILW >RU @ SDUDPHWHUV YDOXHV ZLWK WKH DVVRFLDWHG ILUVW IRXU PRPHQWV DUH JLYHQ LQ 7DEOH*RRGILWLVDFKLHYHGWKRXJKLWLVZRUVHWKDQWKDWREWDLQHGIRU FRPSDUHWRUHVXOWVLQ7DEOH 2IIVSULQJ,17,,, $VVXPLQJ ' WKLV WLPH ZLWK UHVSHFW WR WKH RULJLQ ,17 ZH REWDLQZLWKUHGHILQHGSDUDPHWHUV \ 0 H[S^ Β& > $] &±@`]!$ 8QOLNH,17,DQG,,WKLV,17SUHVHUYHVWKHQRUPDODQGORJQRUPDODV H[DFWVSHFLDOFDVHV)RUWKHIRUPHUFDVHLQVHUWLQ % & )RU WKH ODWWHU FDVH ORJQRUPDO DVVXPH & $OWHUQDWLYHO\ DVVXPH WKDW $] WRREWDLQ \ 0 H[S^ Β& >H[S &$] ±@` D )XUWKHUDVVXPH& WRREWDLQDORJQRUPDOTXDQWLOH
\ 0 H[S $Β]
E
)LWWLQJ ,17 ,,, VLPLODUO\ WR ,17 ,, LQ WKH SUHYLRXV VXEVHFWLRQ WKH UHVXOWLQJSDUDPHWHUVDQGPRPHQWVDUHGLVSOD\HGLQ7DEOH1RWHWKDW WKH SDUDPHWHU $ LV RFFDVLRQDOO\ ODUJH 7KLV LPSOLHV WKDW FDXWLRQ PXVW EH H[HUFLVHG QRW WR XVH LQ WKH ILWWHG ,17 ,,, YDOXHV RI ] VPDOOHU WKDQ$
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH 3DUDPHWHUV YDOXHV DQG WKH ILUVW IRXU PRPHQWV DVVRFLDWHG ZLWK ILWWHG ,17 H[DFWPRPHQWVDUHXSSHUHQWULHV ,17,, XSSHUHQWULHV \ ] 0 H[S^ Α >H[S %] @` ,17,,, PLGGOHHQWULHV \ ] 0 H[S^ Β& > $] &@` ,17,9 ORZHUHQWULHV \ ] 0 H[S>$%] $] @ 'LVWULEXWLRQ 1RUPDO µ σ *DPPD α β *DPPD α β *DPPD α β :HLEXOO α β :HLEXOO α β :HLEXOO α β 5D\OHLJK σ ([WUHPH 9DOXH α β
$
%
&
0HDQ
9DU
6N
.X
&+$37(5,19(56(1250$/,=,1*75$16)250$7,216
2IIVSULQJ,17,9 ([DPLQLQJWKHSDUDPHWHUV YDOXHVREWDLQHGIRU,17,,, 7DEOH RQH QRWHV WKDW & LV QHDU IRU PDQ\ RI WKH H[KLELWHG GLVWULEXWLRQV QRWZLWKVWDQGLQJ WKHLU ZLGHO\ YDU\LQJ VNHZQHVV )RU H[DPSOH IRU WKH JDPPD ZLWK 6N ZH REWDLQ & )RU :HLEXOO ZLWK 6N WKHYDOXHRI&LV 7KLVHPSLULFDOREVHUYDWLRQVXJJHVWV UHGXFLQJWKHQXPEHURISDUDPHWHUVLQ,17,,, E\VHWWLQJ& 7KH UHVXOWLQJ,17,9LV \ 0 H[S>$%] $] @ )LWWLQJ,17,9VLPLODUO\WRWKH,17VLQWKHSUHYLRXVVXEVHFWLRQVWKH UHVXOWLQJSDUDPHWHUVDQGPRPHQWVDUHGLVSOD\HGLQ7DEOH1RWHWKDW WKH QRUPDO GLVWULEXWLRQ LV ZHOO UHSUHVHQWHG DOWKRXJK XQOLNH ,17 ,,, ,17 ,9 GRHV QRW UHGXFH QDWXUDOO\ WR WKH QRUPDO FDVH $OVR QRWH WKDWHYHQWKRXJK KDVRQO\WKUHHSDUDPHWHUVWKHNXUWRVLVPHDVXUH LVDSSUR[LPDWHO\SUHVHUYHGIRUPRVWRIWKHGLVWULEXWLRQVLQ7DEOH )RU DOO WKH SDUDPHWHUUHGXFHG ,17V DOO ILUVW IRXU PRPHQWV DUH SUHVHUYHG WR DQ DFFHSWDEOH GHJUHH QRWZLWKVWDQGLQJ WKH DOJHEUDLF VLPSOLFLW\RIWKHQHZ,17V 'LVWULEXWLRQ)LWWLQJ3URFHGXUHV 0RPHQWEDVHGILWWLQJSURFHGXUHVIRUWKHRIIVSULQJ,17VDUHGHYHORSHG ,W LV DVVXPHG WKDW LQ LPSOHPHQWLQJ WKH ILWWLQJ SURFHGXUH RQO\ VDPSOH HVWLPDWHVRIPRPHQWVDUHDYDLODEOH WKHGLVWULEXWLRQIRUZKLFKZHZLVKWR ILW DQ ,17 LV XQNQRZQ RU XQVSHFLILHG 7KHUHIRUH WKH XQGHUO\LQJ SULQFLSOH LQ GHULYLQJ WKHVH SURFHGXUHV LV WR HPSOR\ RQO\ ORZGHJUHH PRPHQWV VHFRQG GHJUHH DW PRVW DYHUWLQJ WKHUHE\ WKH KLJK VDPSOLQJ LQDFFXUDFLHV DVVRFLDWHG ZLWK VDPSOH HVWLPDWHV RI KLJKGHJUHH PRPHQWV 7KHQHZSURFHGXUHVZLOOODWHUVHUYHWRGHYHORSJHQHUDO63&VFKHPHVIRU QRQQRUPDOHQYLURQPHQWV 6HFWLRQ
5(63216(02'(/,1*0(7+2'2/2*<
)LWWLQJSURFHGXUHVIRU,17, 6HFWLRQ , )LWWLQJZLWKWKHPHGLDQWKHPHDQRI:DQGWKHSDUWLDOPHDQVRI< 5HIHUWR6XEVHFWLRQ ,, )LWWLQJZLWKWKHPHDQDQGYDULDQFHRI:DQGSDUWLDOPHDQVRI< 5HIHUWR6XEVHFWLRQ $ILWWLQJSURFHGXUHIRU,17,, 6HFWLRQ )LWWLQJZLWKWKH PHDQDQGWKHYDULDQFHRI:DQGWKHPHDQRI< $Q H[SUHVVLRQ IRU WKH YDULDQFH RI WKH ORJWUDQVIRUPHG ,17 , ZDV GHYHORSHGLQ6XEVHFWLRQ6LQFH,17,, LVGHULYHGIURP,17 , E\ VHWWLQJ LQ WKH ODWWHU ' ZH REWDLQ IRU WKH YDULDQFH RI : PRGHOHGE\,17,,>UHIHUWR @ 9DU : $>H[S % H[S % @
)URPWKLVZHREWDLQ $ δ^9DU : >H[S % H[S % @`
ZKHUHδ IRU% DQGδ IRU%≥7KLVGHILQLWLRQRIδLVQHFHVVDU\WR HQVXUHWKDW\LQ LVDPRQRWRQHIXQFWLRQRI],QWURGXFLQJIRU$LQWR DQGDOVRIRUORJ 0 >UHIHUWR @
ORJ 0 µ : ±$>H[S % ±@
ZH REWDLQ DQ ,17 LQ RQH SDUDPHWHU % WKDW PD\ EH LGHQWLILHG E\ PDWFKLQJWKHPHDQV 5HVXOWV IURP WKLV ILWWLQJ SURFHGXUH DUH VKRZQ LQ 7DEOH 1RWH WKDW : GHULYHG IURP ,17 ,, LV D OLQHDU WUDQVIRUPDWLRQ RI D ORJQRUPDO YDULDEOH $Q\ FXUUHQW SDUDPHWHUHVWLPDWLRQ SURFHGXUH IRU D ORJQRUPDO YDULDEOHPD\EHHPSOR\HGWRILQGDQHVWLPDWHIRU%
&+$37(5,19(56(1250$/,=,1*75$16)250$7,216
7DEOH 3DUDPHWHU YDOXHV DQG WKH ILUVW IRXU PRPHQWV DVVRFLDWHG ZLWK PRPHQWILWWHG ,17 ,, 6HFWLRQ 0DWFKLQJ E\ PHDQ DQG YDULDQFH RI : DQG E\ PHDQ RI < 0 XQVSHFLILHG8SSHUHQWULHVDUHWKHH[DFWPRPHQWV 'LV 1RUPDO µ σ *DPPD α β *DPPD α β *DPPD α β :HLEXOO α β :HLEXOO α β :HLEXOO α β 5D\OHLJK σ ([WUHPH9DOXH α β
$
%
0HDQ
9DU
6N
.X
$ILWWLQJSURFHGXUHIRU,17,,, 6HFWLRQ )LWWLQJE\WKH PHGLDQWKHPHDQRI:DQGWKHPHDQDQGYDULDQFHRI< 7DNLQJORJRIERWKVLGHVRI DQGWKHQDSSO\LQJWKHH[SHFWHGYDOXH RSHUDWRUWKHPHDQRIWKHORJWUDQVIRUPHGUHVSRQVHLV
µ : ORJ 0 % (^> $= &@&`
'HYHORSLQJ WKH H[SUHVVLRQ LQVLGH WKH H[SHFWHG YDOXH RSHUDWRU ( LQWR D 7D\ORU VHULHV LQ WHUPV RI = DURXQG ]HUR DQG WKHQ WDNLQJ H[SHFWDWLRQIRUWKHILUVWVL[WHUPV QRWHWKDWDOORGGRUGHUPRPHQWVRI= DUHLGHQWLFDOO\]HUR ZHREWDLQ (^> $= &@&`≅ $ & $ & & & ,QWURGXFLQJEDFNLQWR DQGWKHQFHIRU%LQWR ZHREWDLQ ,17,,,H[SUHVVHGLQWHUPVRI0µ : DQGWKHSDUDPHWHUV$DQG&
5(63216(02'(/,1*0(7+2'2/2*<
7KHVHSDUDPHWHUVPD\QRZEHLGHQWLILHGE\ PDWFKLQJWKH PHDQDQG WKH YDULDQFH RI ,17 ,,, ZLWK WKRVH RI < 7DEOH H[KLELWV WKH WUDQVIRUPDWLRQ¶V SDUDPHWHUV DQG WKH UHVXOWLQJ VNHZQHVV DQG NXUWRVLV YDOXHV,QWHUPVRISUHVHUYDWLRQRIDOOILUVWIRXUPRPHQWVDFFHSWDEOHILWLV DFKLHYHG )RU DOO FDVHV WKH YDOXH RI & LV SUDFWLFDOO\ LQFOXGLQJ GLVWULEXWLRQVZKLFKDUHQHDUO\V\PPHWULFDO OLNH5D\OHLJK 7KLVILQGLQJ VXSSRUWVWKHDVVXPSWLRQXVHGWRGHULYH,17,9 )LWWLQJSURFHGXUHVIRU,17,9 6HFWLRQ , )LWWLQJE\WKHPHGLDQ 0 DQGWKHPHDQDQGYDULDQFHRI: 7DNLQJORJRIERWKVLGHVRI ZHREWDLQ :ORJ 0 $%] $]
'HYHORSLQJ WKH 5+6 LQWR D 7D\ORU VHULHV DURXQG ]HUR DQG WDNLQJ H[SHFWDWLRQ >LQWURGXFH & LQWR @ ZH KDYH IRU WKH PHDQ RI WKH ORJWUDQVIRUPHGUHVSRQVH µ : ORJ 0 % $±$
/LNHZLVHVTXDULQJERWKVLGHVRI GHYHORSLQJWKH5+6LQWRD 7D\ORUVHULHVDQGWDNLQJH[SHFWDWLRQJLYH
µ : ±>ORJ 0 @µ : >ORJ 0 @ % $ $
ZKHUH µ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
&+$37(5,19(56(1250$/,=,1*75$16)250$7,216
7DEOH 3DUDPHWHU YDOXHV DQG WKH ILUVW IRXU PRPHQWV DVVRFLDWHG ZLWK PRPHQWILWWHG ,17V WRSWRERWWRP ,17,,, 6HF 0DWFKLQJZLWKWKHPHDQRI:DQGPHDQDQG YDULDQFHRI< 0VSHFLILHG ,17,9 6HF, 0DWFKLQJZLWKWKHPHDQDQGYDULDQFH RI : 0 VSHFLILHG ,17 ,9 6HF ,, 0DWFKLQJ ZLWK WKH PHDQV RI : DQG < 0 VSHFLILHG 'LVWULEXWLRQ 1RUPDO µ σ *DPPD α β *DPPD α β *DPPD α β :HLEXOO α β :HLEXOO α β :HLEXOO α β 5D\OHLJK σ ([WUHPH9DOXH α β
$ QD QD QD QD
% QD QD QD QD
& QD QD
0HDQ
9DU
6N
.X
5(63216(02'(/,1*0(7+2'2/2*<
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`
)URP,17,,, ] $ ^> &% ORJ \0 @ & ±`
)URP,17,9 ] $ ORJ \0 >%ORJ \0 @
:H OHDYH LW DV DQ H[HUFLVH WR ILW WKHVH 17V WR WKH VDPSOHVHW RI GLVWULEXWLRQV XVHG WR ILW WKH ,17V DQG WR HYDOXDWH WKH UHODWLYH HIIHFWLYHQHVV RI WKHVH 17V 1RWH WKDW ILWWLQJ E\ D TXDQWLOHEDVHG1//6SURFHGXUHLVH[SHFWHGWRGHOLYHUDQH[SUHVVLRQIRU WKH UDQGRP YDULDEOH = LQ WHUPV RI < &DOFXODWLQJ QXPHULFDOO\ WKH PRPHQWVRIWKHILWWHG=XVLQJIRUWKHVHFDOFXODWLRQWKHGIRIWKHRULJLQDO UHVSRQVH < VKRXOG UHVXOW LQ PRPHQWV FORVH WR WKRVH RI WKH VWDQGDUG QRUPDOGLVWULEXWLRQ
&+$37(5,19(56(1250$/,=,1*75$16)250$7,216
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
This page intentionally left blank
&KDSWHU
3LHFH:LVH/LQHDU$SSUR[LPDWLRQV
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
5(63216(02'(/,1*0(7+2'2/2*<
OLWHUDWXUH2QHH[DPSOHLVWKH&RUQLVK)LVKHUH[SDQVLRQ 6WXDUWDQG2UG 6HFWLRQ 7KLVH[SDQVLRQSURYLGHVWKHTXDQWLOHRIDUYZLWK NQRZQFXPXODQWVLQWHUPVRIWKHVHFXPXODQWVDQGDSRZHUVHULHVRIWKH FRUUHVSRQGLQJQRUPDOTXDQWLOH [S]S O O ]S O]S O ]S]S O ]S]S «
,Q WKLV HTXDWLRQ ^]S [S` DUH WKH UHVSHFWLYH TXDQWLOH YDOXHV RI WKH VWDQGDUGQRUPDOYDULDEOH=DQGRIWKHVWDQGDUGL]HGUYZKLFKZHZLVK WRUHSUHVHQWE\WKH&RUQLVK)LVKHUH[SDQVLRQ ; <µ σ$OVROULVWKH VWDQGDUGL]HGUWKFXPXODQWRI<
ON κUσU
ZKHUH κU LV WKH UWK XQVWDQGDUGL]HG FXPXODQW DQG σ LV WKH VWDQGDUG GHYLDWLRQ&XPXODQWVDUHIXQFWLRQVRIPRPHQWVRI<DERXWDQDUELWUDU\ SRLQW ^µU ` $ GHWDLOHG H[SODQDWLRQ RI WKHLU GHULYDWLRQ DQG FRPPRQ DSSOLFDWLRQV PD\ EH IRXQG LQ 6WXDUW DQG 2UG 6HFWLRQ ,Q SDUWLFXODU DVVXPLQJ WKDW κ µ QDPHO\ PRPHQWV DUH FDOFXODWHG DERXWWKH PHDQ LW FDQ EH VKRZQ WKDW WKHH[SUHVVLRQV IRUWKH ILUVW IRXU XQVWDQGDUGL]HG FXPXODQWV LQ WHUPV RI FHQWUDO PRPHQWV PRPHQWV DERXWWKHPHDQ DUH WKHUHLQS
κ µ1, κ µ
κ µ3, κ µ4 − µ22,
ZKHUHµULVWKHUWKFHQWUDOPRPHQW IRUH[DPSOHµ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
&+$37(53,(&(:,6(/,1($5$3352;,0$7,216
YHU\VPDOOQXPEHURIWHUPVLVLQFOXGHGLQWKHH[SDQVLRQNQRZOHGJHRI WKH VNHZQHVV DQG NXUWRVLV PHDVXUHV LV UHTXLUHG +RZHYHU VDPSOH HVWLPDWHVRIWKHVHPRPHQWVDUHDVVRFLDWHGZLWKODUJHVDPSOLQJHUURUVDV GLVFXVVHGLQ6HFWLRQ 7KH OLQHDU SLHFHZLVH QRUPDO DSSUR[LPDWLRQV GHYHORSHG LQ WKLV FKDSWHUDUHLQWHQGHGWRDYRLGWKHGUDZEDFNVRIFXUUHQWQRUPDORUQRUPDO EDVHG DSSUR[LPDWLRQV 7KH SLHFHZLVH DSSUR[LPDWLRQV ZHUH LQLWLDOO\ GHYHORSHGLQWKHPLGHLJKWLHVRIWKHODVWFHQWXU\ 6KRUH D 6HYHUDO YDULDWLRQVRIWKHRULJLQDOSUHVHQWDWLRQKDYHVLQFHEHHQGHYHORSHG WKHVH ZLOOEHDOOXGHGWRZLWKUHIHUHQFHVODWHURQ :LWKWKHUHFHQWLQWURGXFWLRQ RI500WKHVHDSSUR[LPDWLRQVQRZVHHPOLNHDQDWXUDOGHULYDWLYHRIWKH 500HUURUGLVWULEXWLRQDQGWKXVFDQEHSHUFHLYHGDVSDUWDQGSDUFHORI WKH500PRGHO $ XQLTXH IHDWXUH RI WKH QHZ DSSUR[LPDWLRQV LV WKHLU FDSDELOLW\ WR SUHVHUYH DOO ILUVW WKUHH PRPHQWV RI WKH GLVWULEXWLRQ WR ZKLFK WKH\ DUH ILWWHG QDPHO\ WKH PHDQ WKH YDULDQFH DQG WKH VNHZQHVV PHDVXUH $V ZLOOEHH[SRXQGHGODWHUWKLVLVDGLUHFWUHVXOWRIWKHXVHRIDSLHFHZLVH OLQHDU DSSUR[LPDWLRQ )XUWKHUPRUH TXDQWLOHV RWKHU WKDQ WKH VWDQGDUG QRUPDO FDQ EH XVHG LQ WKH DSSUR[LPDWLRQV ,Q IDFW TXDQWLOHV RI DQ\ V\PPHWULFDOO\ GLVWULEXWHG UDQGRP YDULDEOH PD\ UHSODFH WKH VWDQGDUG QRUPDOTXDQWLOHV7KLVH[SODLQVZK\LQWKHWLWOHRIWKLVFKDSWHUZHUHIHU WR WKH QHZ DSSUR[LPDWLRQV DV 3LHFHZLVH /LQHDU $SSUR[LPDWLRQV RPLWWLQJ DQ\ PHQWLRQ RI QRUPDO DV DQ HVVHQWLDO HOHPHQW LQ WKHVH DSSUR[LPDWLRQV ,Q WKH QH[W 6HFWLRQ WKH UDWLRQDO IRU WKH SLHFHZLVH OLQHDU DSSUR[LPDWLRQV LV LQWURGXFHG :KHQ D QRUPDO YDULDEOH LV XVHG LQ WKHVH DSSUR[LPDWLRQV WKH\ ZLOO EH GHQRWHG IRU EUHYLW\ WKH PRGLILHG QRUPDO DSSUR[LPDWLRQV ,Q VXEVHTXHQW VHFWLRQV ZH ZLOO GHYHORS LQ GHWDLO YDULDWLRQV RI WKH EDVLF DSSUR[LPDWLRQ DQG WKH DOOLHG ILWWLQJ SURFHGXUHV )RU UHDVRQV VLPLODU WR WKRVH JLYHQ LQ GHYHORSLQJ ILWWLQJ SURFHGXUHV IRU WKHLQYHUVHQRUPDOL]LQJWUDQVIRUPDWLRQV &KDSWHU PRPHQWPDWFKLQJ SURFHGXUHV ZLOO EH FRQILQHG WR PRPHQWV RU WKHLU HVWLPDWHV RI VHFRQG GHJUHHDWPRVW
5(63216(02'(/,1*0(7+2'2/2*<
7KH%DVLF0RGLILHG 1RUPDO $SSUR[LPDWLRQ &RQVLGHUDSORWRIWKHTXDQWLOHIXQFWLRQRIDQ\VNHZHGUDQGRPYDULDEOH DVDIXQFWLRQRIWKHFRUUHVSRQGLQJVWDQGDUGQRUPDOTXDQWLOH6HYHUDOVXFK SORWVZHUHJLYHQLQ&KDSWHU ZKHUHLWZDVGHPRQVWUDWHGKRZZHOOWKH 500 HUURU TXDQWLOHIXQFWLRQ UHSUHVHQWV TXDQWLOHV RI FRPPRQO\ XVHG VNHZHGGLVWULEXWLRQV:HUHLQWURGXFHLQ)LJXUHRQHVXFKSORW7KH SORW UHODWHV WR WKH JDPPD GLVWULEXWLRQ ZLWK SDUDPHWHUV α β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
)LJXUH4XDQWLOHYDOXHVRI ]X VWDQGDUGQRUPDOVWDQGDUGL]HGJDPPD
&+$37(53,(&(:,6(/,1($5$3352;,0$7,216
$JRRGGLYLVLRQSRLQWLVWRKDYHDVHSDUDWHOLQHDUWUDQVIRUPDWLRQIRU ] DQG IRU ]≥ /DWHU RQ WKH VSHFLDO EHQHILWV RI WKLV GLYLVLRQ ZLOO EH GLVFXVVHG $FFRUGLQJO\WKHEDVLFPRGLILHGQRUPDODSSUR[LPDWLRQLV $ ] $ % ] 5# X \µ)σ $ ] $ % 5# ] 7 6
ZKHUH X LV WKH VWDQGDUGL]HG TXDQWLOH RI < DQG ^]\` UHSUHVHQW WKH 3WK TXDQWLOHV RI = WKH VWDQGDUG QRUPDO YDULDEOH DQG RI < QDPHO\ ) \ Φ ] 3DQG)DQGΦDUHWKH&')VRI<DQG=UHVSHFWLYHO\ )LWWLQJ SURFHGXUHV IRU WKLV DSSUR[LPDWLRQ RU YDULDWLRQV WKHUHRI DUH GHYHORSHGLQ6HFWLRQV $9DULDWLRQRIWKH%DVLF0RGHOZLWKD)LWWLQJ3URFHGXUH &RQVLGHUUHZULWLQJWKHEDVLFPRGHODV
#$#! #%& ' & * ] K& -5# ] 6#
\ #$#! #%& ' $ & * ] K& -5# ] 7 6 ZKHUH^K$&`DUHSDUDPHWHUVWKDWQHHGWREHGHWHUPLQHGDQG^µ,σ`DUH WKHPHDQDQGWKHYDULDQFHRI< 7KLVDSSUR[LPDWLRQZDVILUVWLQWURGXFHGLQ6HFWLRQZKHUHFXUUHQW IDPLOLHVRIGLVWULEXWLRQVXVHGLQHPSLULFDOPRGHOLQJKDGEHHQUHYLHZHG 7KHVWUXFWXUHRI LPSOLHVWKDWZHKDYHDVVXPHGXQLIRUPFRQYH[LW\ WKHILUVWSURSHUW\ VLQFHWKHVORSHIRU]≥LVDOZD\VODUJHUWKDQWKDWIRU ] DVVXPLQJ &! ,I < LV D GLVFUHWH UY WKH WHUP σ VKRXOG EH DGGHGWRERWKSDUWVRQWKHULJKWKDQGVLGHWRDFFRXQWIRUWKHIDFWWKDWD TXDQWLOH RI D GLVFUHWH YDULDEOH < LV DSSUR[LPDWHG E\ D FRQWLQXRXV YDULDEOH = 9DOXHV IRU WKH SDUDPHWHUV ^K$&` DUH VRXJKW ZKLFK SUHVHUYH LQ DSSUR[LPDWLRQ WKHILUVWWKUHHPRPHQWVRI<RUHTXLYDOHQWO\WKH ILUVWWKUHHPRPHQWVRIWKHVWDQGDUGL]HGYDULDEOH8 <µ σ.
358
RESPONSE MODELING METHODOLOGY
A modified approximation for the quatile, u, of U, is [from (21.5)] [(Α − C ) z − hC ], z < 0 u= [(Α + C ) z − hC ], z ≥ 0,
(21.5a)
Let us develop the first three moments of (21.5a). This requires knowledge of the partial moments of Z. Discarding the assumption that Z is normal, however assuming that it is a standardized symmetrically distributed variable, denote by MU(Zr) the r-th upper partial moment of Z, namely, MU(Zr) =
∫
∞
0
z r φ ( z )dz .
(21.6)
The upper partial mean is MU(Z). The lower partial mean is ML(Z)= µMU(Z). Also, since Z is symmetrical, MU(Z2)= ML(Z2)=1/2. For simplicity we will henceforth denote MU(Zr) by Mr. Due to the symmetry of the distribution of Z, for all odd-order moments, the sum of the upper and the lower r-th partial moments is zero (the lower partial moment is the negative of the upper partial moment). Similarly, even-order upper and lower partial moments (of the same order) are equal. Values of partial moments for various Z (like the standard normal or the standard logistic) will be provided later on. It can easily be shown that for h=2M1, (21.5a) has zero mean, irrespective of the values of the other parameters. To obtain a unit variance, we need to have for (21.5a) Var(U) = 1 = A2 + C2(1-4M12). (21.7) To preserve the skewness measure, we equate the given Sk (the skewness of U) with the skewness calculated from (21.5a) (refer for details to Shore, 1986a): Sk = 6C(M3 - M1) + 2C3(12M12M3 - 4M13 - 2M3). (21.8) A solution, {A,C}, for the set of equations, (21.7) and (21.8), plus the value assigned to h earlier, ensures that (21.5a) preserves all first three moments of U. Since (21.8) depends on C only, and since this is a thirddegree polynomial, we may find one explicit solution for C (the other two solutions include imaginary values). Re-writing (21.8) as
&+$37(53,(&(:,6(/,1($5$3352;,0$7,216
N&N&±6N
D
ZHREWDLQIRUWKHUHDOYDOXHGVROXWLRQ & ^ N>N6N NN6N @`
^ N> N6N NN6N @`
,QWURGXFLQJEDFNLQWR ZHFDQVROYHIRU$SURYLGHGWKDW
& 0 ≥
7KH H[SUHVVLRQ LQ LQGLFDWHV WKDW SUHVHUYDWLRQ RI VNHZQHVV LV SRVVLEOHE\WKHDOJHEUDLFVWUXFWXUHDVVXPHGLQ D HYHQWKRXJKPHUH OLQHDU WUDQVIRUPDWLRQV RI = DUH XVHG 2IFRXUVH UHJXODU FRQGLWLRQV WKDW HQVXUHH[LVWHQFHRIDUHDOYDOXHGURRWIRUDWKLUGGHJUHHSRO\QRPLDOKDYH WREHVDWLVILHG &RQVLGHUVRPHFDQGLGDWHVIRU= IRUDOO0 &DVH=LV6WDQGDUG1RUPDO )RUDVWDQGDUGQRUPDO=ZHKDYH
0 π 0 π
:HREWDLQIRU DQG UHVSHFWLYHO\ $ &
,QWURGXFLQJ N DQG N LQWR & FDQ EH H[SUHVVHG LQ WHUPV RI 6N 7KH SDUDPHWHU $ PD\ WKHQFH EH HDVLO\ FDOFXODWHGIURP &DVH=LV6WDQGDUG/RJLVWLF 7KH VWDQGDUGL]HG ORJLVWLF YDULDEOH LV GHILQHG LQ 6HFWLRQ )RU WKLV FDVHZHKDYH 0 0
6N &&
,QWURGXFLQJLQWR DQG UHVXOWVLQ $ &
6N & &
5(63216(02'(/,1*0(7+2'2/2*<
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Φ ] ZKHUHΦ LVWKHVWDQGDUGQRUPDO&')WKUHHRI WKHDSSUR[LPDWLRQVDUH ] ^>3 3 @ `]≥
] ^ 3 3ORJ> 3 3@`]≥
] ORJ> 3 3@]≥
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
&+$37(53,(&(:,6(/,1($5$3352;,0$7,216
7DEOH)LUVWIRXUXSSHUSDUWLDOPRPHQWVRIDSSUR[LPDWLRQVWRWKHLQYHUVHGLVWULEXWLRQ IXQFWLRQRIWKHVWDQGDUGQRUPDOGLVWULEXWLRQ 6RXUFH 6WDQGDUG1RUPDO H[DFW (T (T (T
0 π
0
0 π
0
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`
7KH SDUDPHWHUV ^$&` GHWHUPLQHG E\ SUHVHUYH WKH PHDQ DQG WKH YDULDQFH RI WKH DSSUR[LPDWHG GLVWULEXWLRQ +RZHYHU WKH q DFWXDO VNHZQHVV RI GHQRWH LW E\ 6N LV IURP WKH 5+6 RI DIWHULQWURGXFLQJIRU&IURP
q
6N 6N> 6N 00 @ 0000 &RQVLGHUWKHFDVHVH[DPLQHGHDUOLHU
5(63216(02'(/,1*0(7+2'2/2*<
&DVH=LV6WDQGDUG1RUPDO )URP UHVSHFWLYHO\ & 6N 6N
$ 6N
6N 6N6N
q
&DVH=LV6WDQGDUG/RJLVWLF
& 6N 6N
$ 6N
6N 6N6N
q
&DVH6KRUH V$SSUR[LPDWLRQVWRWKH6WDQGDUG1RUPDO ([SUHVVLRQVREWDLQHGDUHVLPLODUWRWKRVHREWDLQHGLQ 7KHUHDGHU PD\ZLVKWRGHULYHWKHVHH[SUHVVLRQVEDVHGRQWKHDSSUR[LPDWLRQVJLYHQ LQ q :H UHDOL]H IURP WKH H[SUHVVLRQV IRU 6N WKDW SURYLGHG WKH DFWXDO VNHZQHVV RI WKH PRQLWRULQJ VWDWLVWLF LV QRW H[WUHPHO\ KLJK 6N LV ZHOO SUHVHUYHG E\ WKH DSSUR[LPDWH TXDQWLOH IXQFWLRQ JLYHQ LQ ZLWK SDUDPHWHUVJLYHQE\ DQG $OVRLI6NLVQRWYHU\KLJKIRU DOO =V ZH KDYH $≈ ,QWURGXFLQJ $ PD\ IXUWKHU VLPSOLI\ WKH XVH RI DSSUR[LPDWLRQ :HFRQFOXGHWKLVVHFWLRQZLWKDQREVHUYDWLRQUHJDUGLQJWKHDOJHEUDLF VWUXFWXUHRI 2EVHUYLQJWKHSDUDPHWHUV&DQG$JLYHQLQ DQG UHVSHFWLYHO\ RQH UHDOL]HV WKDW WKHUH LV D FORVH DVVRFLDWLRQ EHWZHHQWKHVLJQRI6NDQGWKHFRQYH[LW\RIWKHTXDQWLOHIXQFWLRQRI< >DVDSSUR[LPDWHGE\ @)RUSRVLWLYH6NWKHVORSHRI D IRU]!
CHAPTER 21 PIECE-WISE LINEAR APPROXIMATIONS
363
is larger than that for z<0, implying that the true relationship is uniformly convex (second derivative of the exact unknown expression for the quantile function is positive). Conversely, when Sk is negative, this implies that the quantile function is concave (second derivative is negative). This observation is probably true also for the quantile function of the RMM error distribution, developed in Chapter 9, and for the inverse normalizing transformations, developed in Chapter 20. Note, however, that this observation is not a rigorous mathematical proof for the relationship between Sk and the convexity (concavity) of the actual quantile relationship between the response, Y, and Z. A mathematical proof for the "inverse" observation, namely, that a convex (concave) transformation of a symmetrically distributed random variable results in positive (negative) skewness may be found in Van Zwet (1964). 21.5. A Fitting Procedure Using First- and Second-Degree Moments To describe the two-moment fitting procedure, re-write (21.4) in terms of the quantile of the un-standardized response, Y: A1 z + B1 , z < 0, y= A z + B , z ≥ 0. 2 2
(21.22)
Similarly to the definition of the r-th upper partial moment of Z, Mr, given in (21.6), the r-th upper partial moment of Y, Mr(Y), is defined by Mr(Y) =
∫
∞
M
y r f ( y )dy ,
(21.23)
where f(y) is the density function of Y, and M is its median. Thus, to estimate, say, M2(Y) from sample data, we need to calculate the squared values of all data larger then the median, and then calculate the average of all data points, where values lower than the median are replaced by zero. For example, for the data y={7,2,15,3}, an estimate of M2(Y) would be (72+152)/4= 68.5.
5(63216(02'(/,1*0(7+2'2/2*<
,Q WKLV VHFWLRQ WKH SDUDPHWHUV RI DUH GHWHUPLQHG RU HVWLPDWHG E\DPRPHQWPDWFKLQJSURFHGXUHEDVHGRQPDWFKLQJILUVWDQG VHFRQGGHJUHHPRPHQWVSDUWLDODQGFRPSOHWH )URP WKH SDUWLDO ORZHU DQG XSSHU ILUVW DQG VHFRQGGHJUHH PRPHQWV PD\ EH HDVLO\ FDOFXODWHG LQ WHUPV RI WKH SDUWLDO ILUVW DQG VHFRQGGHJUHHPRPHQWVRI= 0DQG0UHVSHFWLYHO\ DQGLQWHUPVRI WKHSDUDPHWHUVRI (TXDWLQJWKHVHWRWKHFRUUHVSRQGLQJPRPHQWV RI<ZHREWDLQIRUWKHSDUDPHWHUVRI $ ^ σ2 + µ2) 0 < >µ0 < @`> 0@
% >µ0 < $0@
$ ^0 < >0 < @`> 0@
% >0 < $0@
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
&+$37(53,(&(:,6(/,1($5$3352;,0$7,216
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α DQG β= 7KH PHGLDQ 0 DQG WKH UHODWHG ILUVW WKUHH PRPHQWVDUH
0 µ σ2 6N
,WLVHDVLO\YHULILHGE\QXPHULFDOLQWHJUDWLRQWKDWWKHILUVWWZRSDUWLDO PRPHQWVRIWKHJDPPDGLVWULEXWLRQZLWKWKHVSHFLILHGSDUDPHWHUVDUH 0 < 0 < 8VLQJIRU=WKHVWDQGDUGQRUPDOYDULDEOHZHREWDLQIURP IRUWKH SDUDPHWHUVRI K 0 $ &
366
RESPONSE MODELING METHODOLOGY
Similarly, from (21.24), the parameters of (21.22) are A1 = 2.023, B1 = 4.976, A2 = 4.864, B2 = 4.756.
(21.28)
Introducing the parameters in (21.27) into (21.5), a piece-wise linear approximation is obtained for the quantile of Y in terms of the standard normal quantile. From this, the r-th non-central moment of Y is µr' = E(Yr) =
∫
0
−4
[σ ( A − C ) z − 0.7978Cσ + µ ]r φ ( z )dz
4
+
∫ [σ ( A + C ) z − 0.7978Cσ + µ ] φ ( z)dz , r
0
(21.29)
where φ(z) is the density function of the standard normal variable, and the limits of integration were set arbitrarily as z=-4 and z=4. Similarly, introducing the parameters in (21.28) into (21.22), we obtain for the r-th non-central moment of Y: µr' = E(Yr) =
∫
0
−4
[ A1 z + B1 ]r φ ( z )dz
4
+
∫ [ A z + B ] φ ( z )dz 0
r
2
2
(21.30)
From (21.29) and (21.30), the first four moments are µ = µ1', σ2 = µ2' − (µ1')2, Sk = [µ3' - 3µ2' µ1' + 2(µ1')3] / σ3, Ku = [µ4' - 4µ3' µ1' + 6µ2' (µ1')2 - 3(µ1')4] / σ4 - 3
(21.31)
Table 21.2 displays the resulting first four moments. In this table, (21.22) [the un-standardized (21.4)] is fitted by the two-moment fitting procedure, with parameters given by (21.24). Also, (21.5) is fitted by the three-moment procedure, with parameters calculated from (21.20) (namely, Sk is approximately preserved). For both models, the first three moments are well preserved. Note, in particular, that for (21.22) only moments of second degree at most took part in the fitting procedure. However, the skewness measure is preserved.
&+$37(53,(&(:,6(/,1($5$3352;,0$7,216
7DEOH)LUVWIRXUPRPHQWVRI< XVLQJDVWDQGDUGQRUPDO] $SSUR[LPDWLRQ XVHG (T WZRPRPHQW ILWWLQJ (T WKUHHPRPHQW ILWWLQJ ([DFW PRPHQWV
0HDQ
9DULDQFH
6N
.X
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
5(63216(02'(/,1*0(7+2'2/2*<
>@6KRUH+ D $QHZHVWLPDWHRIVNHZQHVVZLWK06(VPDOOHUWKDQWKDWRIWKH VDPSOH VNHZQHVV &RPPXQLFDWLRQV LQ 6WDWLVWLFV 6LPXODWLRQ DQG &RPSXWDWLRQ >@ 6KRUH + E 2SWLPXP VFKHGXOH IRU SUHYHQWLYH PDLQWHQDQFH D JHQHUDO VROXWLRQ IRU D SDUWLDOO\ VSHFLILHG WLPHWRIDLOXUH GLVWULEXWLRQ 3206 $Q ,QWHUQDWLRQDO -RXUQDO RI WKH 3URGXFWLRQ DQG 2SHUDWLRQV 0DQDJHPHQW 6RFLHW\ >@6KRUH+ *HQHUDOFRQWUROFKDUWVIRUDWWULEXWHV,,(7UDQVDFWLRQV >@ 6WXDUW $ 2UG - . .HQGDOO¶V $GYDQFHG 7KHRU\ RI 6WDWLVWLFV 9 'LVWULEXWLRQ7KHRU\&KDUOHV*ULIILQ &R/WG/RQGRQ >@ 9DQ =ZHW : 5 &RQYH[ 7UDQVIRUPDWLRQV RI 5DQGRP 9DULDEOHV 0DWKHPDWLVFK&HQWUXP$PVWHUGDP
&KDSWHU
*HQHUDO&RQWURO&KDUWV
,QWURGXFWLRQ 7KH500RULJLQDOPRGHODQGVRPHRILWVGHULYDWLYHV OLNHWKHSLHFHZLVH OLQHDUDSSUR[LPDWLRQV PD\VHUYHDVJRRGSODWIRUPVWRFRQVWUXFWFRQWURO VFKHPHV IRU VWDWLVWLFDO SURFHVV FRQWURO 63& ,Q 6HFWLRQ JHQHUDO SURFHVV FRQWURO VFKHPHV IRU DWWULEXWHV DUH GHYHORSHG DQG OLNHZLVH IRU YDULDEOHVLQ6HFWLRQ 7KH WHUP *HQHUDO LQ WKH WLWOH RI WKLV FKDSWHU QHHGV TXDOLILFDWLRQ 7UDGLWLRQDO 6KHZKDUW FKDUWV DVVXPH DSSUR[LPDWH QRUPDOLW\ IRU WKH PRQLWRULQJVWDWLVWLFKHQFHWKH±FRHIILFLHQWVLQWKHH[SUHVVLRQVIRUWKH FRQWUROOLPLWV7KHQHZFRQWUROFKDUWVH[WHQGWKHVSHFWUXPRIVFHQDULRV ZKHUH63&PD\EHLPSOHPHQWHGWRQRQQRUPDOHQYLURQPHQWV RUZKHQ DSSUR[LPDWH QRUPDOLW\ FDQQRW EH DVVXPHG :H XVH SLHFHZLVH OLQHDU DSSUR[LPDWLRQV &KDSWHU WR GHULYH JHQHUDO FRQWURO FKDUWV IRU DWWULEXWHVDQGZHXVHLQYHUVHQRUPDOL]LQJWUDQVIRUPDWLRQV &KDSWHU WRGHYHORSJHQHUDOFRQWUROFKDUWVIRUYDULDEOHV 7KURXJKRXW WKLV FKDSWHU FRQWURO OLPLWV UHIHU WR WKH WUDGLWLRQDO 6KHZKDUWOLPLWV WKH±σOLPLWV 3UREDELOLW\OLPLWVUHODWHWRRWKHUOLPLWV GHWHUPLQHG E\ VRPH VSHFLILHG UHTXLUHG IDOVH DODUP SUREDELOLWLHV WKH SUREDELOLWLHV RI H[FHHGLQJ WKH OLPLWV ZKHQ WKH SURFHVV LV LQ D VWDWHRI FRQWURO *RRG VRXUFHV IRU 63& DUH 0RQWJRPHU\ DQG *UDQW DQG /HDYHQZRUWK
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5*(1(5$/&21752/&+$576
/LNHZLVH YDULRXV VROXWLRQV KDYH EHHQVXJJHVWHGIRUVFHQDULRV ZKHQ WKHWUDGLWLRQDO6KHZKDUWFKDUWVIRUDWWULEXWHVFDQQRWEHLPSOHPHQWHGGXH WRH[WUHPHVNHZQHVV RIWKHGLVWULEXWLRQRIWKHVWDWLVWLFXVHGWRPRQLWRU WKH SURFHVV ([DPSOHV DUH WKH DUFVLQ WUDQVIRUPDWLRQ DSSOLHG WR ELQRPLDO GDWD WKH XVH RI WKH ³4 FKDUW´ IRU PRQLWRULQJ ELQRPLDO DQG 3RLVVRQ SDUDPHWHUV 4XHVHQEHUU\ DE RU WKH XVH RI WKH ³JFKDUW´ DQGWKH³KFKDUW´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
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5*(1(5$/&21752/&+$576
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µ VWDQGDUGGHYLDWLRQ σ) DQGVNHZQHVV 6N WKHODWWHULVWKHWUDGLWLRQDOPHDVXUHRIVNHZQHVV QDPHO\6N µσDQGµLVWKHWKLUGFHQWUDOPRPHQW 'HQRWHE\TDQG T TT WKHUHTXLUHG)DOVH$ODUP5DWHV )$5V ZLWKUHVSHFWWR WKH ORZHU SUREDELOLW\ OLPLW /3/ DQG WR WKH XSSHU SUREDELOLW\ OLPLW 83/ UHVSHFWLYHO\7KLVLPSOLHVWKDWIRUDVWDEOHSURFHVVZHUHTXLUHWKDW
/7$ 3U < /3/ ≤T87$ 3U
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
5(63216(02'(/,1*0(7+2'2/2*<
DVKRXOGEHDGGHGWRERWKSUREDELOLW\OLPLWVDQGDXQLW VKRXOG EHDGGHGWRWKH/3/ DVDOOXGHGWRHDUOLHU $FFRUGLQJO\DQGJLYHQWKHVSHFLILHGµσDQG6NWKHH[SUHVVLRQVIRU WKHSUREDELOLW\OLPLWVRIDJHQHUDODWWULEXWHFRQWUROFKDUWDUH
83/ µ]T $& σ &σ
&/ µ
/3/ µ]T $& σ &σ + ,
ZKHUH$DQG&DUHWKHVROXWLRQWR &DVHLQ6HFWLRQ $ &6N &&
)RU WKH VHFRQG H[SUHVVLRQ LQ H[WUHPH YDOXHV DUH REWDLQHG DW & ± ZKHUH 6N ± 7KLV LPSOLHV WKDW D VROXWLRQ IRU ^$ &` PD\ EH REWDLQHG LQ WKH UDQJH 6N ^ ` )RU PRVW FXUUHQW 6KHZKDUW DWWULEXWHV FKDUWV WKLV UDQJH LV VXIILFLHQW ,I ZH ZLVK WR H[WHQG WKHDSSOLFDELOLW\RIWKHQHZFRQWUROOLPLWVWRFDVHVZLWKKLJKHUVNHZQHVV OLNHWKHJHRPHWULFGLVWULEXWLRQZLWKDVNHZQHVVYDOXHQHDU ZHPD\ XVHWKHVWDQGDUGL]HGORJLVWLFTXDQWLOH &DVHLQ6HFWLRQ ]S √π ORJ>3 3 @ ZKHUH 3 LV WKH YDOXH RI WKH &') 5HODWLYH WR WKH QRUPDO YDULDEOH WKH ORJLVWLFKDVORQJHUWDLOV7KHUHIRUHLWVTXDQWLOHVPD\EHPRUHDSSURSULDWH LQFRQVWUXFWLQJFRQWUROOLPLWVIRUKLJKO\VNHZHGGLVWULEXWLRQV 3UREDELOLW\ OLPLWV FRUUHVSRQGLQJ WR DQG PD\ HDVLO\ EH GHYHORSHG7KHUHVXOWLQJH[SUHVVLRQVDUH
83/ µ]T $& σ&σ
&/ µ
/3/ µ]T $& σ&σ + ,
ZKHUH$DQG&DUHWKHVROXWLRQWR>(T @ $ &6N & &
)RU WKH VHFRQG H[SUHVVLRQ LQ H[WUHPH YDOXHV DUH REWDLQHG DW & ± ZKHUH 6N ± 7KLV LPSOLHV WKDW D VROXWLRQ IRU ^$ &` PD\ EH REWDLQHG LQ WKH UDQJH 6N ^ ` 7KLV LV D ZLGHU UDQJH WKDQWKDWDVVRFLDWHGZLWKWKHQRUPDO $QRWKHUEHQHILWRIXVLQJWKHORJLVWLFLVWKDWUHODWLYHWRWKHQRUPDOWKH
&+$37(5*(1(5$/&21752/&+$576
DSSUR[LPDWH VLPSOLILHG H[SUHVVLRQV IRU $ DQG & GHYHORSHG LQ 6HFWLRQ DQGXVHGLQ6XEVHFWLRQ EHWWHUSUHVHUYHWKHVNHZQHVVRIWKH XQGHUO\LQJGLVWULEXWLRQ>FRPSDUHH[SUHVVLRQVIRU6NLQ QRUPDO ZLWK ORJLVWLF@ (PSOR\LQJ WUDGLWLRQDO )$5V DVVRFLDWHG ZLWK 6KHZKDUW FKDUWV QDPHO\ T T ZH REWDLQ IURP IRU WKH FRQWURO OLPLWV EDVHGRQVWDQGDUGQRUPDOTXDQWLOHV LQWURGXFH] ±IRU]
8&/ µ $& σ &σ
&/ µ
/&/ µ $& σ &σ + ,
ZKHUH$DQG&DUHGHWHUPLQHGDVEHIRUH )RUWKHORJLVWLFWKHFRUUHVSRQGLQJHTXDWLRQVDUH LQWURGXFH] ± IRU]
8&/ µ $& σ&σ
&/ µ
/&/ µ $& σ&σ + ,
ZKHUH$DQG&DUHGHWHUPLQHGDVEHIRUH 6LPSOLILHG OLPLWV ZLOO EH LQWURGXFHG LQ WKH QH[W 6XEVHFWLRQ :H ZLOO FRQILQH RXUVHOYHV WR WKH QRUPDO FDVH ZKLFK UHQGHUV WKH QHZ DSSURDFK D QDWXUDO H[WHQVLRQ RI WKH H[LVWLQJ 6KHZKDUWFKDUWV $OVR WKH QRUPDO FDVH ZLOO EH PRUH VXLWDEOH IRU DWWULEXWHV WKDW DUH DSSUR[LPDWHO\ QRUPDOO\GLVWULEXWHGOLNHWKHELQRPLDODQGWKH3RLVVRQ)RUYHU\KLJKO\ VNHZHGGLVWULEXWLRQVRUGLVWULEXWLRQVWKDWGRQRWFRQYHUJHWRWKHQRUPDO OLNHWKHJHRPHWULF XVLQJFRQWUROOLPLWVEDVHGRQORJLVWLFTXDQWLOHVPD\ SURYHWREHPRUHHIIHFWLYH 6LPSOLILHGOLPLWV 7KH QHHG WR VROYH QXPHULFDOO\ IRU $ DQG & >UHIHU WR @ PD\ EH FXPEHUVRPH6LPSOHUOLPLWVFDQEHGHULYHG IRUHLWKHUSUREDELOLW\OLPLWV RU FRQWURO OLPLWV ZKLFK FLUFXPYHQW WKLV GLIILFXOW\ $OVR WKH VLPSOLILHG H[SUHVVLRQVDOORZH[SUHVVLQJWKHOLPLWVH[SOLFLWO\LQWHUPVRIWKHPHDQ WKHVWDQGDUGGHYLDWLRQDQGWKHVNHZQHVVPHDVXUH
5(63216(02'(/,1*0(7+2'2/2*<
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µ]Tσ> 6N 6N@ σ6N &/ µ,
/3/ µ]Tσ > 6N 6N@ σ6N )RU6N≤WKHVNHZQHVVRIWKHDSSUR[LPDWHTXDQWLOHIXQFWLRQXVHG WR FRQVWUXFW WKH SUREDELOLW\ OLPLWVLQ GRHV QRWGHYLDWH PRUH WKDQ ± IURP WKH H[DFW YDOXH >UHIHU WR @ *LYHQ WKDW WKH XQGHUO\LQJ PRQLWRULQJ VWDWLVWLF LV DQ DWWULEXWH DQG WKHUHIRUH DVVXPHV RQO\LQWHJHUYDOXHV WKLVGHYLDWLRQLVRIOLWWOHFRQVHTXHQFH ,QWURGXFLQJ] ±LQWR WKHFRQWUROOLPLWVDUH
8&/ µσ 6N σ6N
&/ µ,
/&/ µσ 6N σ6N
$ IXUWKHU VLPSOLILFDWLRQ PD\ EH LQWURGXFHG E\ REVHUYLQJ WKDW WKH VTXDUHURRW LQ DQG >WKH SDUDPHWHU $ LQ @ LV YHU\ FORVHWR IRUQRQH[WUHPHYDOXHVRI6N 6XEVWLWXWLQJWKLVYDOXHIRUWKH VTXDUHURRWLQ ZHREWDLQVLPSOLILHGSUREDELOLW\OLPLWV
83/ µ]Tσ + σ6N ]T
&/ µ,
/3/ µ]Tσ −σ6N ]T 6LPLODUO\IRUWKHFRQWUROOLPLWV
&+$37(5*(1(5$/&21752/&+$576
8&/ µσ σ6N
&/ µ,
/&/ µσ σ6N
8QOLNHHTXDWLRQV DQG GHULYHGIURPDTXDQWLOHIXQFWLRQ WKDW SUHVHUYHV H[DFWO\ WKH PHDQ DQG WKH VWDQGDUG GHYLDWLRQ RI WKH PRQLWRULQJ VWDWLVWLF WKH ODVW WZR HTXDWLRQV DUH EDVHG RQ D TXDQWLOH IXQFWLRQWKDWSUHVHUYHVRQO\WKHPHDQZKLOHWKHVWDQGDUGGHYLDWLRQDQG WKH VNHZQHVV DUH RQO\ DSSUR[LPDWHO\SUHVHUYHG)RU6N≤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
5(63216(02'(/,1*0(7+2'2/2*<
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`REWDLQHGH[KLELWDKLJKGHJUHHRI FRQVLVWHQF\ LUUHVSHFWLYH RI WKH XQGHUO\LQJ DWWULEXWH GLVWULEXWLRQ )XUWKHUPRUH QHDUO\ HTXDO YDOXHV DUH REWDLQHG IRU / DQG IRU / 6RPH YDOXHV DUH VKRZQ LQ 7DEOH IRU GHILQLWLRQV RI SDUDPHWHUV UHIHU WR 6XEVHFWLRQ *LYHQ WKH FORVH SUR[LPLW\ EHWZHHQ WKH YDOXHV RI / DQG / LW LV GHFLGHGWRVHWHTXDOYDOXHVIRUERWKFRHIILFLHQWVDW/ / $FFRUGLQJO\WKHILQDOSUREDELOLW\OLPLWVDIWHUVNHZQHVVLQIODWLRQDUH
83/ µ]Tσ + ( σ6N ]T
&/ µ,
/3/ µ]Tσ − σ6N ]T
6LPLODUO\IRUWKHFRQWUROOLPLWVDIWHUVNHZQHVVLQIODWLRQZHREWDLQ
8&/ µσσ6N
&/ µ
/&/ µσσ6N
7R DVVHVV WKH SHUIRUPDQFH RI WKHVH FRQWURO OLPLWV WKH DVVRFLDWHG $5/VQHHGWREHFDOFXODWHG $5/LVDFRPPRQO\XVHGPHDVXUHIRUWKH VHQVLWLYLW\RIWKHFRQWUROFKDUWWRWKHH[LVWHQFHRIRXWRIFRQWUROVWDWHV 7KLVZLOOEHGRQHLQ6XEVHFWLRQZKHUHWKHUHODWLYHHIIHFWLYHQHVV RIWKHQHZFRQWUROOLPLWVLVDVVHVVHG
&+$37(5*(1(5$/&21752/&+$576
7DEOH6RPHYDOXHVRI/ IRUWKH/3/ DQGRI/ IRUWKH83/
'LVWULEXWLRQ 3RLVVRQ>λ @ %LQRPLDO>Q @ S S S *HRPHWULF >S @ 1HJDWLYH%LQRPLDO > NS ^ `@
/
/
3UREDELOLW\OLPLWVIRUVRPHDWWULEXWHGLVWULEXWLRQV $SSO\LQJWKHJHQHUDOH[SUHVVLRQVGHYHORSHGLQWKHSUHYLRXVVXEVHFWLRQ 6KHZKDUWOLNH FRQWURO OLPLWV PD\ EH FRQVWUXFWHG IRU VRPH VSHFLILHG GLVWULEXWLRQV0RGLILHGFRQWUROOLPLWVZLOOEHGHYHORSHGIRUWKHWUDGLWLRQDO QSDQGFFRQWUROFKDUWVDQGQHZOLPLWVZLOOEHGHYHORSHGIRUDWWULEXWHV ZLWK JHRPHWULF DQG QHJDWLYHELQRPLDO GLVWULEXWLRQV )RU WKH ODWWHU WKH SUREDELOLW\ IXQFWLRQV XVHG LQ GHULYLQJ WKH PRPHQWVDQG WKH SUREDELOLW\ OLPLWVDUHUHVSHFWLYHO\
I \ S S \\
I \
\ $ N N S S \\ \
1RWH WKDW WKH QHJDWLYHELQRPLDO UDQGRP YDULDEOH GHVFULEHV WKH QXPE HU RI ³)DLOXUHV´ < EHIRUH WKH RFFXUUHQFH RI WKH NWK ³6XFFHVV´ LQ UHSHDWHGO\SHUIRUPHGLQGHSHQGHQW%HUQRXOOLWULDOV WULDOVWKDWUHVXOWLQ RU ZLWKVXFFHVVSUREDELOLW\S 7KH UHVXOWV DUH VXPPDUL]HG LQ 7DEOH 6LQFH WKH SUREDELOLW\ IXQFWLRQRIWKHJHRPHWULFGLVWULEXWLRQLVDPRQRWRQHGHFUHDVLQJIXQFWLRQ RI \ WKHUH LV QR PHDQLQJ WR GHILQLQJ DQ /3/ IRU WKLV GLVWULEXWLRQ +RZHYHU D 83/ REYLRXVO\ PDNHV VHQVH VLQFH LW ZLOO HQDEOH GHWHFWLQJ LPSUREDEOH KLJK YDOXHV RI < WKH PRQLWRULQJ VWDWLVWLF 1RWH WKDW DOWKRXJKSUREDELOLW\ FRQWURO OLPLWVEDVHGRQQRUPDOTXDQWLOHVGHOLYHU
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH3UREDELOLW\OLPLWVIRUWKHELQRPLDO3RLVVRQJHRPHWULFDQGQHJDWLYH ELQRPLDO 'LVWULEXWLRQ
3UREDELOLW\/LPLWVPLQXVFHQWHUOLQH /3/µ 83/µ ]T>QS S @ ]T>QS S @ S ± S %LQRPLDO QS QS S S σ [ ]T [ ]T± ± ]T λ ± ]T λ 3RLVVRQ λ λ λ [ ]T [ ]T± ± ]T> S S@ > S S@ S S *HRPHWULF S S S S җ [ ]T± ± ]T>N S S@ ]T>N S S@ S 1HJDWLYH N S N S > S S@ ±> S S@ >N S @ [ ]T %LQRPLDO S S [ ]T± ± µ
0RPHQWV σ2 6N
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
&+$37(5*(1(5$/&21752/&+$576
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
5(63216(02'(/,1*0(7+2'2/2*<
&RPSDULVRQ PD\ QRZ EH FRQGXFWHG ZLWK WKH H[DFW RSWLPDO FRQWURO OLPLWV>GHULYHGIURP EDVHGRQWKHH[DFW/7$DQG87$@6RPH YDOXHVREWDLQHGIURPYDULRXVGLVWULEXWLRQVDUHGLVSOD\HGLQ7DEOH 7DEOH&RPSDULVRQRIH[DFWZLWKDSSUR[LPDWHRSWLPDO&/VIRUWKHELQRPLDOWKH 3RLVVRQWKHJHRPHWULFDQGWKHQHJDWLYHELQRPLDOGLVWULEXWLRQV7KHQRUPDO DSSUR[LPDWLRQLVDOVRJLYHQ ZLWKFRQWLQXLW\FRUUHFWLRQ XQGHU6KHZKDUW 'LVWU %LQRP QS 3RLVλ *HRPS 1HJ%LQ NS
([DFW&/ /&/ 8&/
= H[DFW =/ =8
1HZ&/ /&/ 8&/
6KHZKDUW&/ /&/ 8&/
&+$37(5*(1(5$/&21752/&+$576
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`DQGQ H[DFW RSWLPDO FRQWUROOLPLWVDFFRUGLQJ WRWKH /)FULWHULRQ KDYH EHHQ FDOFXODWHG 7KHVH DUH FRPSDUHG WR YDOXHV REWDLQHG IURP XVLQJ VLPSOHURXQGLQJ DQGZLWKWKHFODVVLFDO6KHZKDUWFRQWUROOLPLWV ZLWKWKH DGGHGFRQWLQXLW\FRUUHFWLRQDJDLQXVLQJVLPSOHURXQGLQJ 7KHGLVWULEXWLRQRIWKHGHYLDWLRQVIURPWKHRSWLPDOOLPLWVLVJLYHQLQ 7DEOH 7KH QHZ FRQWURO OLPLWV IRU WKH ELQRPLDO RXWSHUIRUP WKH WUDGLWLRQDOOLPLWVHVSHFLDOO\IRUVPDOOSYDOXHV 7DEOH'LVWULEXWLRQ LQ RIGHYLDWLRQVIURPH[DFWRSWLPDOFRQWUROOLPLWV)RUHDFK JLYHQSFDVHVZHUHH[DPLQHG>Q @/RZHUHQWULHVUHIHUWR6KHZKDUW FKDUWV ([DFW $SSUR[ ]HURGHY S /&/ 8&/
([DFW≠$SSUR[ GHYRI± /&/
8&/
([DFW≠$SSUR[ _GHY_! /&/
8&/
5(63216(02'(/,1*0(7+2'2/2*<
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
&+$37(5*(1(5$/&21752/&+$576
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µ+3σIRUWKHXSSHU
5(63216(02'(/,1*0(7+2'2/2*<
FRQWURO OLPLW 8&/ DQG µ−3σ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α,β`7KHILUVWIRXUPRPHQWVDUH
µ = αβ, σ2 = αβ26N α.X α.
&+$37(5*(1(5$/&21752/&+$576
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
µ
αβ, σ2
αβ 2Q6N αQ .X αQ
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
5(63216(02'(/,1*0(7+2'2/2*<
GHWHUPLQH WKH GLVWULEXWLRQ RI WKH VDPSOH DYHUDJH WKH PRQLWRULQJ VWDWLVWLF ,IDFHUWDLQ,17LVXVHGWRUHSUHVHQWDSURFHVVGLVWULEXWLRQPRQLWRULQJ WKH SDUDPHWHUV RI WKLV ,17 LPSOLHV PRQLWRULQJ WKH SDUDPHWHUV RI WKH SURFHVV GLVWULEXWLRQ WKDW WRRN SDUW LQ ILWWLQJ WKH ,17 7R UHDOL]H WKDW VXSSRVH WKDW D FHUWDLQ ILWWLQJ SURFHGXUH IRU WKH VHOHFWHG ,17 UHTXLUHV VSHFLILFDWLRQRIWKHUHVSRQVHPHGLDQWKHUHVSRQVHPHDQDQGWKHUHVSRQVH YDULDQFH UHODWHWRWKHILWWLQJSURFHGXUHVGHOLQHDWHGLQ6HFWLRQ 7KLV LPSOLHV WKDW LQ RUGHU WR HQVXUH WKH VWDELOLW\ RI WKH ,17 DQG E\ LPSOLFDWLRQ WKH VWDELOLW\ RI WKH SURFHVV GLVWULEXWLRQ UHSUHVHQWHG E\ WKH ,17 RQO\WKHVHUHVSRQVHSDUDPHWHUVQHHGWREHPRQLWRUHG 7ZR63&VFKHPHVDUHVXJJHVWHGKHUH7KHVFKHPHVGLIIHUDFFRUGLQJ WRWKH,17XVHGDQGDFFRUGLQJWRWKHDOOLHGILWWLQJSURFHGXUH7KHODWWHU GHWHUPLQHV ZKLFK PRPHQWV RU RWKHU HVWLPDEOH SDUDPHWHUV RI WKH UHVSRQVH GLVWULEXWLRQ QHHG WR EH PRQLWRUHG 'DWD UHTXLUHPHQWV DUH DOVR GHWHUPLQHGE\WKHUHOHYDQWILWWLQJSURFHGXUH 7KHVFKHPHVGLVSOD\HGDUHQRWWKHRQO\RQHVWKDWRQHFDQFRQFHLYHRI EDVHG RQ WKH ,17V 2WKHU VFKHPHV SHUKDSV ZLWK EHWWHU SHUIRUPDQFH SURSHUWLHVPD\EHFRQVWUXFWHG7KHIROORZLQJVFKHPHVVKRXOGWKHUHIRUH EH UHJDUGHGMXVWDV GHPRQVWUDWLRQ RI KRZ WKH ,17V PD\ EH XVHG LQWKH LPSOHPHQWDWLRQRI63&WRQRQQRUPDOSURFHVVGLVWULEXWLRQV 6FKHPH,'DWDUHTXLUHPHQWV (VWLPDWHVRIΜ, µ DQGµ : /HW < EH D SURFHVV YDULDEOH WR EH PRQLWRUHG DQG OHW : EH WKH ORJ WUDQVIRUPHG UHVSRQVH QDPHO\ : ORJ < ,Q &KDSWHU ZH KDYH LQWURGXFHGDWKUHHSDUDPHWHU,17UHLQWURGXFHGKHUHIRUFRQYHQLHQFH : ORJ < ORJ 0 $%= $= ZKHUH0LVWKHPHGLDQDQG=LVDVWDQGDUGQRUPDOYDULDEOH'HYHORSLQJ WKH ULJKWKDQGVLGH LQ WHUPV RI = LQWR D 7D\ORU VHULHV DQG WDNLQJ H[SHFWDWLRQRIWKHILUVWILYHWHUPVZHREWDLQ
µ : ORJ 0 % $$
,QWURGXFLQJ IRU % EDFN LQWR : LV H[SUHVVHG LQ WHUPV RI WKH PHGLDQWKHPHDQµ : DQGWKHSDUDPHWHU$
: ORJ 0 >µ : ORJ 0 @$=> $$ $= @
&+$37(5*(1(5$/&21752/&+$576
7KH SDUDPHWHU $ PD\ QRZ EH GHWHUPLQHG E\ PDWFKLQJ WKH UHVSRQVH PHDQ µ)ZLWK(>H[S : @DVWKHODWWHULVLPSOLHGIURP )URP WKLV PDWFKLQJ SURFHGXUH LW LV DSSDUHQW WKDW ,17 LV XQLTXHO\ GHWHUPLQHG E\ 0 µ : DQG µ (VWLPDWHV RI WKHVH SDUDPHWHUV ZRXOG EH XVHG DVWKH PRQLWRULQJ VWDWLVWLFV WR HQVXUH WKH VWDELOLW\ RI WKH SURFHVV GLVWULEXWLRQ 7KH UHVSHFWLYH FRQWURO VFKHPH LV UHIHUUHG WR DV 6FKHPH, $OWHUQDWLYHO\GHYHORS:LQWRD7D\ORUH[SDQVLRQDURXQGµDQGWDNH H[SHFWDWLRQRIWKHILUVWWKUHHWHUPVWRREWDLQ WKHVHFRQGWHUPYDQLVKHV
µ : ≅ORJ µ σ/µ)2
0RQLWRULQJµ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µ DQG σ ZLWK DQ ,17 GLIIHUHQWIURP @LVJLYHQODWHULQWKLVVHFWLRQ UHIHUWR6FKHPH,, 7KH IROORZLQJ SURFHGXUH UHODWHG WR 6FKHPH , LV EDVHG RQ WKH DVVXPSWLRQ WKDW ,17 LV ILWWHG YLD Μ, µ DQG µ : 7KHUHIRUH WKHVHDUHWKHSDUDPHWHUVWKDWQHHGWREHPRQLWRUHG,WLVDOVRDVVXPHGWKDW HVWLPDWHV RI WKHVH SDUDPHWHUV KDYH GLVWULEXWLRQV WKDW FDQ EH DGHTXDWHO\ GHVFULEHGE\,17 7KHSURFHGXUHEHORZLVGLYLGHGLQWRWZRSDUWV
3KDVH )LWWLQJ HVWLPDWLQJ WKH,17VDQGFDOFXODWLQJFRQWUROOLPLWV 3KDVH ,PSOHPHQWLQJWKHFRQWUROVFKHPH
5(63216(02'(/,1*0(7+2'2/2*<
(VWLPDWLQJWKH,17VDQGFDOFXODWLQJWKHUHVSHFWLYHFRQWUROOLPLWV 'UDZ.LQGHSHQGHQWVDPSOHVRIQREVHUYDWLRQVHDFK&DOFXODWHIRUWKH MWKVDPSOH M . WKHPHGLDQ0MWKHPHDQµMDQGWKHPHDQRIWKH ORJRIWKHRULJLQDOREVHUYDWLRQVµM : *LYHQ WKH . YDOXHV ^0M` HVWLPDWH WKHLU PHGLDQ WKHLU PHDQ DQG WKH PHDQ RI WKH ORJWUDQVIRUPHG VDPSOHPHGLDQV 5HSHDW WKH VDPH FDOFXODWLRQV IRU WKH VDPSOHPHDQ >EDVHG RQ ^µM`@ DQG IRU WKH VDPSOH PHDQRIWKHORJWUDQVIRUPHGREVHUYDWLRQV>EDVHGRQ^µM : `@ )LW ,17 IRU WKH GLVWULEXWLRQV RI WKH WKUHH VWDWLVWLFV XVHG WR PRQLWRU WKH SURFHVV QDPHO\ WKH VDPSOH PHGLDQ WKH VDPSOH PHDQ DQG WKHVDPSOHPHDQRI: ,QWKHILWWLQJSURFHGXUHVXVHVDPSOHHVWLPDWHV GHULYHG IURP WKH SUHOLPLQDU\ VHW RI . VDPSOHV DV H[SODLQHG LQ WKH SUHYLRXVSDUDJUDSK )URP WKH WKUHH HVWLPDWHG TXDQWLOH IXQFWLRQV ILQG FRQWURO OLPLWV WR PRQLWRU0µDQGµ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
&+$37(5*(1(5$/&21752/&+$576
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σ VLQFH XVLQJ WKH VWDQGDUG GHYLDWLRQ GRHV QRW SDUWLFLSDWHLQGHWHUPLQLQJWKHSDUDPHWHUV¶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
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH)RXUVHWVRIFRQWUROOLPLWVIRUDQDYHUDJHFRQWUROFKDUW 6DPSOHVL]HQ 'LV *DPPD α β α β α β α β α β α β α β
6HW$ H[DFW /&/ 8&/
6HW% 6HW& 6HW' (T 6KHZKDUWH[σ 6KHZKDUWDSσ җ /&/ 8&/ /&/ 8&/ /&/ 8&/
FRUURERUDWHVRXUDVVHUWLRQWKDWXVLQJ6FKHPH,SURYLGHVDOOWKHUHTXLUHG LQIRUPDWLRQ DERXW WKH SURFHVV GLVWULEXWLRQ DQG QR PRQLWRULQJ RI YDULDELOLW\LVQHHGHG 7R DVVHVV WKH SHUIRUPDQFH RI WKH ; FKDUW ZKHUH WKH OLPLWV DUH GHWHUPLQHGEDVHGRQWKHVSHFLILHGPHGLDQPHDQDQGPHDQRIWKHORJRI WKHVDPSOHDYHUDJH FRUUHVSRQGLQJWR6HW%LQ7DEOH WKHSDUDPHWHU αRIWKHJDPPDSURFHVVGLVWULEXWLRQLVFKDQJHGWR δ αZKHUH δ ^` (DFKFKDQJHδ,LQFXUVDFKDQJHLQWKHSURFHVVPHDQIURPµWR δ µ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α≤ 6KHZKDUW ORZHUFRQWUROOLPLWLVQHJDWLYH7KHUHIRUHRFFDVLRQDOO\QR/&/YDOXHVDUH JLYHQIRUWKHVHYDOXHVRIα ([DPLQLQJ7DEOHDIHZREVHUYDWLRQVDUHQRWHZRUWK\
&+$37(5*(1(5$/&21752/&+$576
7DEOH $5/V DVVRFLDWHG ZLWK GLIIHUHQW GHYLDWLRQV δ LQ WKH SURFHVV PHDQ ZKHQ FRQWURO OLPLWV XVHG DUH 6HW % UHIHU WR 7DEOH ([ UHIHUV WR WKH H[DFW $5/V $SUHIHUVWRWKHDSSUR[LPDWH$5/V FDOFXODWHGIURPWKHILWWHG,17XVLQJSRVWFKDQJH SDUDPHWHUV YDOXHV 6KHZUHIHUVWR6KHZKDUW;FKDUWZKHUHFRQWUROOLPLWVDUHGHILQHG E\6HW& UHIHUWR7DEOH 7KH$5/VDUHWKHH[DFWYDOXHV 'LVWULEXWLRQ &RQWURO δ= OLPLWV *DPPD β /&/ /&/ 8&/ α ([ $S 6KHZ α ([ $S 6KHZ ( ( α ([ $S 6KHZ ( ( ( α ([ $S 6KHZ α ([ $S 6KHZ α ([ $S 6KHZ α ([ $S 6KHZ
8&/
)LUVWWKH$5/VFRPSXWHGIURPWKHILWWHG,17DUHQHDUO\LGHQWLFDOWR WKRVH FDOFXODWHG IURP WKH H[DFW SRVWFKDQJH DYHUDJHGLVWULEXWLRQ 7KLV LPSOLHV WKDWLIWKH XQGHUO\LQJ GLVWULEXWLRQLV QRW NQRZQ WKH $5/V PD\ VWLOO EH FDOFXODWHG LI JRRG HVWLPDWHV IRU WKH GLVWULEXWLRQDO SDUDPHWHUV QHHGHGWRILW DUHDYDLODEOH $VHFRQGREVHUYDWLRQUHIHUVWRWKHPDOSHUIRUPDQFHRIWKHWUDGLWLRQDO 6KHZKDUWFKDUWYLVDYLVIDOVHDODUPUDWHV7KH$5/VLQDVWDWHRIFRQWURO GHYLDWH FRQVLGHUDEO\ IURP WKH VWDQGDUG QRPLQDO YDOXH RI FRPSDUDEOH UHVXOWV DUH UHSRUWHG LQ %RUURU HW DO 2Q WKH RWKHU
5(63216(02'(/,1*0(7+2'2/2*<
KDQG6(7%FRQWUROOLPLWVFDOFXODWHGZLWKQRNQRZOHGJHRIWKHSURFHVV YDULDQFHGHOLYHUH[DFW$5/VWKDWDUHFORVHWRWKHGHVLUHGVWDQGDUGYDOXH $ WKLUG REVHUYDWLRQ UHIHUV WR WKH LQDELOLW\ RI WKH 6KHZKDUW FKDUW WR GHWHFWGRZQZDUGVKLIWVLQWKHPHDQIRUα≤&RQYHUVHO\6HW%FRQWURO OLPLWVSURYLGHQRQQHJDWLYHFRQWUROOLPLWVWKDWPD\EHYHU\LQVWUXPHQWDO LQ GHWHFWLQJ ODUJH GRZQZDUG VKLIWV REVHUYH WKH VPDOO $5/V IRU ODUJH QHJDWLYHδ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Μ, µDQGσ ,Q&KDSWHUWKHIROORZLQJIRXUSDUDPHWHU,17ZDVGHYHORSHGDQGLWLV UHLQWURGXFHGKHUHIRUFRQYHQLHQFH : ORJ < ORJ 0 %>H[S &] @'] ZKHUH 0 LV WKH PHGLDQ 7KLV ,17 KDV EHHQ VKRZQ WR EH SDUWLFXODUO\ HIIHFWLYH LQ UHSUHVHQWLQJ TXDQWLOH IXQFWLRQV RI ERWK FRQWLQXRXV DQG GLVFUHWHUDQGRPYDULDEOHV 6KRUHE 7DNLQJH[SHFWDWLRQRI ZHREWDLQ ILQGGHWDLOVLQ&KDSWHU
µ : ORJ 0 %>H[S & @
,QWURGXFLQJIRU%IURP LQWR DQGDVVXPLQJWKDW0LV VSHFLILHGZHREWDLQDTXDQWLOHIXQFWLRQLQWZRSDUDPHWHUV&DQG'7KH ODWWHUPD\QRZEHGHWHUPLQHGE\PDWFKLQJWKHPHDQVDQGWKHYDULDQFHV ZLWKWKRVHRIWKHUHVSRQVHYDULDEOH< 5HFDOOLQJ WKDW PRQLWRULQJ µ : LV DSSUR[LPDWHO\ HTXLYDOHQW WR PRQLWRULQJµDQGσ2 RQHOHDUQVWKDWPRQLWRULQJWKHPHGLDQWKH
&+$37(5*(1(5$/&21752/&+$576
PHDQ DQG WKH YDULDQFH ZRXOG JXDUDQWHH WKH VWDELOLW\ RI WKH SURFHVV GLVWULEXWLRQDVWKHODWWHULVPRGHOHGE\,17 :HOHDYHLWDVDQRSHQH[HUFLVHWRFRQVWUXFWDFRQWUROVFKHPHEDVHG RQ DQGVWXG\LWVSURSHUWLHVIRUDQDVVXPHGJDPPDGLVWULEXWLRQ VLPLODUO\WRWKHDQDO\VLVFRQGXFWHGIRU6FKHPH,DERYH
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λ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
5(63216(02'(/,1*0(7+2'2/2*<
>@6KRUH+ D *HQHUDOFRQWUROFKDUWVIRUDWWULEXWHV,,(7UDQVDFWLRQV >@ 6KRUH + E *HQHUDO FRQWURO FKDUWV IRU YDULDEOHV ,QWHUQDWLRQDO -RXUQDO RI 3URGXFWLRQ5HVHDUFK > @ 6KRUH + 0RGHOLQJ D QRQQRUPDO UHVSRQVH IRU TXDOLW\ LPSURYHPHQW ,QWHUQDWLRQDO-RXUQDORI3URGXFWLRQ5HVHDUFK >@ 6WXDUW $ 2UG - . .HQGDOO¶V $GYDQFHG 7KHRU\ RI 6WDWLVWLFV 9 'LVWULEXWLRQ7KHRU\&KDUOHV*ULIILQ &R/WG/RQGRQ >@:KHHOHU'- 6KHZKDUW VFKDUWV 0\WKVIDFWVDQGFRPSHWLWRUV7UDQVDFWLRQV RIWKH$64&WK$QQXDO4XDOLW\&RQJUHVV >@ :RRGDOO :+ &RQWURO FKDUWLQJ EDVHG RQ DWWULEXWH GDWD %LEOLRJUDSK\ DQG 5HYLHZ-RXUQDORI4XDOLW\7HFKQRORJ\ >@
&KDSWHU
,QYHQWRU\$QDO\VLV
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
5(63216(02'(/,1*0(7+2'2/2*<
)LUVW$SSURDFK7KH4XDQWLOH)XQFWLRQDQGWKH/RVV)XQFWLRQ ,Q&KDSWHUWKHSLHFHZLVHOLQHDUDSSUR[LPDWLRQZDVGHULYHG $ ] $ %] 5# \ $ ] $ % 5# ] 7 ZKHUH\LVWKHTXDQWLOHRIDUDQGRPYDULDEOH UY <KDYLQJGLVWULEXWLRQ IXQFWLRQ ) \ DQG ] LV WKH FRUUHVSRQGLQJ TXDQWLOH RI = D VWDQGDUGL]HG UYZLWKDV\PPHWULFGLVWULEXWLRQ 6N 5HH[SUHVVHGLQDGLIIHUHQWIRUPRQHPD\ZULWHIRU #$#! #%&' & * ] K& -5# ] # \ #$#! #%&' $ & * ] K& -5# ] 7 ZKHUH^K$&`DUHWKHVROXWLRQRI K 0
$ & 0
6N & 00 & 0000
0LLVWKHLWKXSSHUSDUWLDOPRPHQWRI= WKHLWKGHJUHHPRPHQWRIWKH XSSHUKDOIRIWKHGLVWULEXWLRQRI= DQG^µ, σ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
&+$37(5,19(1725<$1$/<6,6
WKHREMHFWLYHIXQFWLRQLQWKHRSWLPL]DWLRQSUREOHP7KH/)LVGHILQHGE\
U
U
X
X
T
T
/< X \ X I \ G\ > ) \ @G\ \≥
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r 6N *G]5#3 r G\ ! $ 6N G] 3 7 )RUWKHVWDQGDUGL]HGORJLVWLF
G] √π >3 3 @G3
ZKHUH3 Φ(] DQGΦLVWKH&')RI= ,QWURGXFLQJ LQWR DQG WKHQFH LQWR WKH ORVV IXQFWLRQ DVVRFLDWHGZLWK ZLWK$ LV 8 ! ^ 6N >ORJ ORJ 3 @ /< 3 6N ORJ `3 8 ! 6N ORJ 3 3 7 ZKHUH3 ) X DQG) LVWKHGLVWULEXWLRQIXQFWLRQRI< )RUPDQ\VWRFKDVWLFRSWLPL]DWLRQPRGHOVERWKWKHTXDQWLOHIXQFWLRQ DQGWKHORVVIXQFWLRQDUHLQWHJUDOFRPSRQHQWVRIWKHREMHFWLYHIXQFWLRQ
5(63216(02'(/,1*0(7+2'2/2*<
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
]S ^ 3 3ORJ> 3 3@`3≥ 7KHDVVRFLDWHGORVVIXQFWLRQLV
&+$37(5,19(1725<$1$/<6,6
^ORJ>3 3 @`3 / = 3 3 3 3 7
,QWURGXFLQJLQWR ZHKDYH / 3 $ ^ORJ>3 3 @` $ $ 3 < $ 3 33 7
6LQFH = XVHG LV DQ DSSUR[LPDWLRQ IRU WKH VWDQGDUG QRUPDO YDULDWH WKH SDUDPHWHUVDUH UHIHUWR6HFWLRQ $ σ 6N % µ − σ6N
$ σ 6N % µ − σ6N
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
402
RESPONSE MODELING METHODOLOGY
In this section, we use for inventory analysis a new four-parameter family of distributions (Shore, 1998), which satisfies these requirements. As was done in the previous section, the quantile function and the loss function are first developed. A maximum likelihood estimation procedure is then derived to estimate the parameters of the new distribution. In Section 23.5 the new family of distributions is applied to obtain an optimal solution for a certain inventory problem. Let Y be the response (a r.v.), and assume that Y is bounded by zero from below, and is unbounded, otherwise. Let F(y) be the cumulative distribution function (CDF), and denote by µ and σ the mean and the standard deviation, respectively. Let y be the P-th quantile, namely, F(y)=P. Consider the following inverse distribution function (the quantile function): A [P/(1-P)]B1 , P < 1/2 y(P) = 1 (23.13) B A 2 {[P/(1-P)] 2 - 1} + A1 , P ≥ 1/2, where {Ai, Bi} (i=1,2) are parameters that need to be determined. The motivation for introducing this quantile function, and some of its characteristics, are treated in Shore (1998). The density function (d.f.) of Y is 1-B 1+B [1/(A1 B1 )]P 1 (1-P) 1 , P<1/2, f[y(P)] = (23.14) 1-B 1+B [1/(A1 B1 )]P 2 (1-P) 2 , P ≥ 1/2. Denote by Mi(Y) the i-th upper partial moment of Y, namely, 1
Mi(Y) =
∫
P =1 / 2
[ y ( P)]i dP ,
(23.15)
with y(P) given by (23.13), and dP= f(y)dy. Note that in (23.15) integration is with respect to P (not with respect to y). This practice will often be repeated where P, rather than y, is used as the variable of integration or differentiation. Pursuing the two-moment (partial and complete) matching approach (refer to Chapter 21 for motivation and details), an estimation procedure may be developed to identify the parameters in (23.13) (refer to Shore, 1995b, 1998). However, the procedure requires a numeric search for the
&+$37(5,19(1725<$1$/<6,6
YDOXHV RI WKH SDUDPHWHUV 7KLV UHQGHUV WKH HVWLPDWLRQ SURFHGXUH VRPHZKDWFXPEHUVRPHIRUSUDFWLWLRQHUV$PRGLILHGSURFHGXUHLVKLJKO\ GHVLUDEOHDQGLWZLOOEHGHYHORSHGODWHURQ 1H[WFRQVLGHU<WKDWLVV\PPHWULFDOO\GLVWULEXWHGDQGXQERXQGHGLQ ERWK GLUHFWLRQV 'HQRWH WKH TXDQWLOH RI WKH VWDQGDUGL]HG YDULDEOH E\ X (\−µ)/σ 7KH IROORZLQJ TXDQWLOH IXQFWLRQ UHSODFHV IRU WKLV FKDUDFWHUL]DWLRQRIWKHUDQJHRIYDULDWLRQRI< $^>3 3 @% `3 X 3 % $^>3 3 @ `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
404
RESPONSE MODELING METHODOLOGY 1
Li(P) =
∫
P
[AiBi][F/(1-F)]Bi(1/F)dF,
(i=1 for P<1/2, i=2, otherwise).
(23.18)
It may easily be shown that the loss function, associated with (23.13), is L (P) + [L 2 (1/2) - L1 (1/2)], P < 1/2, L Y (P) = 1 L 2 (P), P ≥ 1/2.
(23.19)
This expression will be used in subsequent sections. Suppose now that the moments, needed to determine the parameters of (23.13) via moment matching, are unknown, and have to be estimated from sample observations. It has been previously noted (Shore, 1996, 1998) that estimates of the parameters of (23.13) tend to have large sampling variability due to the fact that B1 and B2 appear as exponents. In particular, B2 tends to be close to zero and is particularly susceptible to sampling deviations. An alternative approach that circumvents this difficulty is needed. To develop the modified procedure, let us impose a requirement for continuity of the d.f. at P=1/2. This implies that [refer to (23.14)]: A1B1= A2B2= K, say, and (23.13) becomes (K/B1 )[P/(1-P)]B1 , P < 1/2, y(P) = B (K/B2 ){[P/(1-P)] 2 - 1} + A1 , P ≥ 1/2.
(23.20)
The parameter "B2" is expected to be close to zero (observe representative values in Shore, 1995b). Since, for non-negative u, Lim λ: [(uλ -1)/λ] = log(u), we may rewrite (23.20) as A [P/(1-P)]B1 , P < 1/2, y(P) = 1 A2 log[P/(1-P)] + B2 , P ≥ 1/2.
(23.21)
Two new parameters are introduced for P≥1/2, to allow for a twomoment fitting. The “penalty" to be paid is the loss of continuity of the quantile function at P=1/2. For a two-moment fitting procedure, using partial and complete
&+$37(5,19(1725<$1$/<6,6
PRPHQWV WKH H[SUHVVLRQV IRU WKH SDUDPHWHUV LQ DUH 6KRUH % ^µ 9 >µ 9 @`
$ H[S^>µ 9 %@`
$ ^0 < >0 < @`
% >0 < $@
ZKHUH9LVWKHORJWUDQVIRUPDWLRQ ORJ < < 0 9 < 7 0 µ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
5(63216(02'(/,1*0(7+2'2/2*<
7DEOH3DUDPHWHUVVNHZQHVV 6NDS DQGNXUWRVLV .XDS RIWKHILWWHG 3DUDPHWHUVFDOFXODWHGIURP 7KHH[DFWYDOXHVDUH6NDQG.XUHVSHFWLYHO\ 'LVWULEXWLRQ *DPPD α β 6N .X α β 6N .X /RJQRUPDO µ σ 6N .X µ σ 6N .X :HLEXOO α β 6N .X α β 6N .X α β 6N .X
$
%
$
%
6NDS
.XDS
7KHORVVIXQFWLRQRI LV / 3 >/ / @3 / < 3 $ ORJ 3 3 7 ZLWK/ 3 JLYHQE\ DQG/ 3 $ORJ 3 3≥ 5HIHUDOVRWR 8QWLO WKLV SRLQW WKH SDUDPHWHUV RI WKH TXDQWLOH IXQFWLRQ RI WKH UHVSRQVH < KDYH EHHQ HVWLPDWHG YLD PRPHQW PDWFKLQJ +RZHYHU WKH SDUDPHWHUVLQ PD\EHHDVLO\HVWLPDWHGYLDPD[LPXPOLNHOLKRRG $VVXPHWKDWWKHUHDUHQDYDLODEOHREVHUYDWLRQV^\ N ` N Q ZKHUH \ N LV WKH NWK RUGHUVWDWLVWLF WKHUH DUH N VPDOOHU REVHUYDWLRQV 7KH OLNHOLKRRGIXQFWLRQ / LVIURP
CHAPTER 23 INVENTORY ANALYSIS
L = [1/(A1B1)] (n/2)
∏
n/2 k =1
407
[ P(k) 1-B1 (1-P(k))1+B1 ]
x [1/(A2B2)] (n/2)
∏
n k = n / 2 +1
[ P(k)1-B2 (1-P(k))1+B2],
(23.25)
where P(k)= F[y(k)], and n is assumed to be an even number. Introducing from (23.13) for P(k) in terms of y(k), (23.25) is expressed in terms of the given observations. The maximum-likelihood estimates for {Ai, Bi} (i=1,2) may now be identified by maximizing log(L) either via differentiation or by a numerical search. A similar procedure can be developed for (23.21). Other fitting and estimation procedures, including procedures for right- and left-censored observations, are developed in Shore (1996). The case of truncated observations may be particularly interesting for models with lost sales. The reader is referred to the above reference for the appropriate fitting procedures. 23.4. First Approach- Newsboy Problem with Order-up-to Policy A general solution for the classical newsboy problem is derived, pursuing the first approach (Section 23.2). We assume that only the first three moments of the demand in the period, D, are known. Let S be the order-up-to level, and IC(S) be the expected inventory cost (ordering, holding and penalty) for the period. It is well known that IC(S) is given by IC(S)= cS + h
∫
S
t =0
(S - t)f D (t)dt + p
∫
∞
t =S
(t - S)f D (t)dt , (23.26)
where fD(d) is the d.f. of D, and {c, h, p} are the unit ordering cost, unit holding cost and unit penalty cost, respectively. It is assumed that penalty cost, p, is applied to excessive demand (that which exceeds available stock and thus create a shortage). It may be easily shown that
∫
S
t =0
(S - t)f D (t )dt = LD(P) + S - E(D),
(23.27)
where LD(P) is the value of the loss function of D at its CDF value FD(S)=P, and E(D) is the expected value of D. Introducing from (23.27)
5(63216(02'(/,1*0(7+2'2/2*<
LQWR DQG WKHQ IRU 6 WKH 3WK TXDQWLOH RI ' IURP ZH REWDLQIRU ,& 6 FK $]S% KS /' 3 ± K ( ' ZKHUH]SLVWKH3WKTXDQWLOHRI=DQG/' 3 PD\EHJLYHQE\DQ\RIWKH IRUPXODH LQ 6HFWLRQ 1RWH WKDW LW LV DVVXPHG WKDW IRU WKH RSWLPDO VROXWLRQ 3 ! 7KLV H[SODLQV ZK\ WKH SDUDPHWHUV ^$ %` DSSHDU LQ &RQVLGHU XVLQJ IRU ]S WKH DSSUR[LPDWLRQ IRU WKH VWDQGDUG QRUPDO YDULDEOH JLYHQ LQ 7KH FRUUHVSRQGLQJ ORVV IXQFWLRQ LV JLYHQ E\ DQG WKH SDUDPHWHUV ^$ % E\ 1RWH WKDW ERWK ]S DQG /' 3 DUHQRZH[SUHVVHGLQWHUPVRIDVLQJOHWHUPQDPHO\ 3 37KLV UHTXLUHVGLIIHUHQWLDWLQJRI ZLWKUHVSHFWWR 3 3 DVVXPLQJWKDW 3! UDWKHUWKDQZLWKUHVSHFWWR3 :HREWDLQWKHZHOONQRZQIRUPXOD H[DFW 3 SF SK
RU
3 3 KF SF
ZKHUH3 LVWKHRSWLPDO3 ,QWURGXFLQJ IURP LQWR DQG WKHQ LQWR ZLWK WKH SDUDPHWHUVJLYHQE\ WKHRSWLPDO6LV 6 $] % $ ^ 3 3 ORJ> 3 3 @`% σ' > 6N ' @ ^ KF SF ORJ> KF SF @±`( ' ± >6N ' @σ('
ZKHUH] LVWKHRSWLPDOTXDQWLOHRI= WKDWFRUUHVSRQGVWR3 DQGσ'DQG 6N ' DUH WKH VWDQGDUG GHYLDWLRQ DQG WKH VNHZQHVV PHDVXUH RI WKH GHPDQG'(TXDWLRQ LVDQH[SOLFLWH[SUHVVLRQIRU6 WKHRSWLPDO YDOXHRIWKHGHFLVLRQYDULDEOHLQWHUPVRIWKHFRVWSDUDPHWHUVDQGRIWKH ILUVWWKUHHPRPHQWVRI',I6 LVDVVXPHGWREHGLVFUHWHDYDOXHRI VKRXOGEHVXEWUDFWHGIURPWKH5+6RI 6LPLODUH[SUHVVLRQVPD\ EHGHYHORSHGIRU3 ,QWURGXFLQJ IRU 6 $] % EDFN LQWR DQG VLPLODUO\ IRU /' 3 ,& 6 LV REWDLQHG LQ WHUPV RI WKH FRVWV DQG WKH ILUVW WKUHH
&+$37(5,19(1725<$1$/<6,6
PRPHQWV RI WKH GHPDQG ' 7KLV JHQHUDO VROXWLRQ LV QRW EDVHG RQ WKH QRUPDODVVXPSWLRQ>6N ' ≠@DQGLVYDOLGZKLOHWKHILUVWWKUHHPRPHQWV DUHVSHFLILHG 7KH UHDGHU PD\ ZLVK WR H[DPLQH WKH DFFXUDF\ RI WKLV VROXWLRQ IRU VRPHQXPHULFDOGDWDDQGVSHFLILHGQRQQRUPDOGLVWULEXWLRQV 1XPHULFDO FRPSDULVRQV IRU WKLV PRGHO DQG RWKHUV ZKHUH WKH ILUVW DSSURDFK ZDVXVHGWR REWDLQ VROXWLRQV IRU D GHPDQGGLVWULEXWLRQWKDW LV RQO\SDUWLDOO\VSHFLILHG YLDWKHILUVWWKUHHPRPHQWV FDQEHIRXQGLQWKH DIRUHFLWHGUHIHUHQFHV 6HFRQG$SSURDFK7ZR([DPSOHV 7KH&RQWLQXRXV5HYLHZ 45 0RGHO 7KLVLVWKHPRVWFRPPRQFRQWLQXRXVUHYLHZPRGHO*LYHQLWV³ROG´DJH UHIHU IRU H[DPSOH WR +DGOH\ DQG :KLWLQ LW VWLOO DWWUDFWV UHODWLYHO\ODUJHUHVHDUFKHIIRUW$UHFHQWH[DPSOHLV3ODWW5RELQVRQDQG )UHXQG :H DGGUHVV KHUH WKH DSSUR[LPDWH EDFNRUGHU YHUVLRQ DV WUHDWHGE\/DXDQG/DX 7KH FRQWLQXRXV UHYLHZ 45 PRGHO DVVXPHV WKDW ZKHQ LQYHQWRU\ OHYHO ZKLFKLVFRQWLQXRXVO\PRQLWRUHG JRHVGRZQWR5DQRUGHURIVL]H 4LVLVVXHG/HDGWLPHGHPDQGLVDUY<ZLWK&')) \ 7KHORQJUXQ DYHUDJHFRVWSHUSHULRGLV
& 45 .( ' 4K>45µ]πΕ(' /< 3 4
ZKHUH 4 WKH ORW VL]H DQG 5 WKH UHRUGHUSRLQW DUH GHFLVLRQ YDULDEOHV /< 3 LVWKHORVVIXQFWLRQDW3 ) 5 .LVWKH IL[HG RUGHUFRVWKDQGπ DUH WKH XQLW FDUU\LQJ FRVW SHU SHULRG DQG WKH XQLW VKRUWDJH FRVW UHVSHFWLYHO\( ' LVWKHDYHUDJHGHPDQGSHUSHULRGDQGµ LVWKHDYHUDJH OHDGWLPHGHPDQG DYHUDJHRI< ,QWURGXFLQJ IRU 5 IURP DQG IRU /< 3 IURP DQG GLIIHUHQWLDWLQJ ZLWK UHVSHFW WR 4 DQG WR 3 ZH REWDLQ WKH RSWLPDOLW\ FRQGLWLRQV 4 >( ' .@K^>( ' π@K`/< 3
3 4 K >( ' π@
5(63216(02'(/,1*0(7+2'2/2*<
/ 3 >/ / @3 / < 3 / 3 3 7 )URPWKHVHH[SUHVVLRQVWKHRSWLPDO 4 3 PD\EHLGHQWLILHGDQG WKHQFH 5 IURP 6LQFH LQ PRVW LQYHQWRU\ PRGHOV LW LV H[SHFWHG WKDW 3 ! WKLV LQWHUYDO VKRXOG EH VHDUFKHG ILUVW IRU WKH RSWLPDO VROXWLRQ$OVRQRWHWKDWLQ 4 DQG3 DUHWKHH[DFWH[SUHVVLRQV DQGRQO\/< 3 LVSDUWLFXODUWRWKHQHZVROXWLRQSURFHGXUH $Q DOWHUQDWLYH SURFHGXUH PD\ EH GHULYHG E\ XVLQJ WKH PRGLILHG H[SUHVVLRQV )RU WKH FRPPRQ FDVH RI 3 ≥ RQH REWDLQVWKHVLPSOHURSWLPDOLW\FRQGLWLRQV 4 >( ' .@K^>( ' π@K`>$ORJ 3 @ >( ' .@K^>( ' π@K`^$ORJ> 4 K ( ' π @`3 ≥
3 4 K >( ' π@
2QO\ D URXWLQH URRWILQGLQJ URXWLQH LV QRZ UHTXLUHG WR LGHQWLI\ 4 )XUWKHUGHWDLOVDERXWWKHGHULYDWLRQRIWKLVPRGHOPD\EHIRXQGLQ6KRUH $QXPHULFDOH[DPSOH )RU WKH FRVW DQG GHPDQG SDUDPHWHUV WDNHQ IURP /DX DQG /DX V H[DPSOHV
( ' . K π =
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
&+$37(5,19(1725<$1$/<6,6
7DEOH&RPSDULVRQRIH[DFWVROXWLRQ 4 3 DQGDSSUR[LPDWHVROXWLRQ 4D 3D IRUWKH 45 PRGHO$SSUR[LPDWHVROXWLRQVXVLQJHLWKHU XSSHUHQWU\ RU >ORZHU HQWU\ ZLWK SDUDPHWHUV JLYHQ LQ 7DEOH 7KH ORZHVW WZR HQWULHV IRU WKH ORJQRUPDODQG:HLEXOOFDVHVGLVSOD\VROXWLRQVGHULYHGXQGHUWKHJDPPDDVVXPSWLRQ 'LVWULEXWLRQ
*DPPD α β 6N 0 /RJQRUPDO µ σ 6N 0
:HLEXOO α=2, β= 5 6N 0
. π=5 π 4 4D 4 3 3D 3 BBB BBB
4D 3D BB BBB
. π π 4 4D 4 4D 3 3D 3 3D BB BBB
BB BBB
:HLEXOO ZLWK SDUDPHWHUV DV JLYHQ LQ 7DEOH +RZHYHU ZH KDYH PLVWDNHQO\ LGHQWLILHG WKH XQGHUO\LQJ OHDGWLPH GLVWULEXWLRQ DV JDPPD ZLWK SDUDPHWHUV WKDW SUHVHUYH WKH DFWXDO PHDQ DQG VWDQGDUG GHYLDWLRQ QDPHO\ α µσ β σµ ZKHUH µ DQG σ DUH WKH DFWXDO PHDQ DQG VWDQGDUGGHYLDWLRQRIWKHOHDGWLPHGHPDQGGLVWULEXWLRQ 7KH ORZHVW WZR HQWULHV RI 7DEOH IRU WKH ORJQRUPDO DQG WKH
5(63216(02'(/,1*0(7+2'2/2*<
:HLEXOO FDVHV SUHVHQW WKH H[DFW RSWLPDO VROXWLRQV XQGHU WKLV VFHQDULR 1RWHWKDW3D VWDQGVIRUWKHDFWXDOYDOXHRI3DVVRFLDWHGZLWKWKHRSWLPDO 5 ZKHUHWKHODWWHUKDGEHHQLGHQWLILHGXQGHUWKHJDPPDDVVXPSWLRQ ([DPLQLQJ7DEOHRQHUHDOL]HVWKDWDOWKRXJKERWKWKHPHDQDQG WKH YDULDQFH DUH SUHVHUYHG WKH ZURQJO\ DVVXPHG JDPPD GLVWULEXWLRQ FRQVLVWHQWO\UHVXOWVLQVROXWLRQVWKDWDUHZRUVHWKDQWKRVHREWDLQHGXQGHU WKHQHZDSSURDFK XVLQJDSSUR[LPDWLRQV $OWKRXJKWKHGLIIHUHQFHVPD\ VHHP QHJOLJLEOH LQ WKLV H[DPSOH FXPXODWLYH H[SHULHQFH ZLWK WKH QHZ DSSURDFKVXJJHVWVWKDWWKHUHVXOWVREWDLQHGKHUHDUHUHSUHVHQWDWLYHRIWKH PRUH JHQHUDO FDVH QDPHO\ WKDW D ZURQJO\ VHOHFWHG GLVWULEXWLRQ WKDW SUHVHUYHV WKH H[DFW ILUVW WZR PRPHQWV JHQHUDOO\ \LHOGV ZRUVH UHVXOWV WKDQ WKRVH GHULYHG E\ WKH QHZ DSSURDFK ZKHUH VNHZQHVV LV SUHVHUYHG 1XPHULFDOHYLGHQFHPD\EHIRXQGLQ6KRUH $OWHUQDWLYHO\ZHPD\ZLVKWRXVHVRPHRWKHUIRXUSDUDPHWHUIDPLOLHV RI GLVWULEXWLRQV OLNH 3HDUVRQ V ZKLFK ZRXOG EH ILWWHG E\ IRXUPRPHQW PDWFKLQJ :KHQ WKH H[DFW PRPHQWV DUH XQNQRZQ DQG RQO\ VDPSOH HVWLPDWHV DUH DYDLODEOH WKLV DSSURDFK ZRXOG DOVR SURYLGH OHVV DFFXUDWH VROXWLRQV UHODWLYHWRWKHQHZDSSURDFK GXHWRWKHODUJHVDPSOLQJHUURUV DVVRFLDWHGZLWKHVWLPDWHVRIVNHZQHVVDQGNXUWRVLV UHODWHWRDGLVFXVVLRQ RIWKLVSUREOHPLQ6HFWLRQ 7R VXP XS UHODWLYH WR WKH WZR RWKHU RSWLRQV DYDLODEOH WR WKH SUDFWLWLRQHUQDPHO\LGHQWLI\LQJWKHOHDGWLPHGHPDQGGLVWULEXWLRQ DQG ULVNLQJPLVLGHQWLILFDWLRQ RUXVLQJVRPHIOH[LEO\VKDSHGIRXUSDUDPHWHU GLVWULEXWLRQ WR EH HVWLPDWHG E\ IRXUPRPHQW PDWFKLQJ WKH QHZ DSSURDFKVHHPVWRFRQVLVWHQWO\RXWSHUIRUPIRUFDVHVZKHUHVDPSOHGDWD DUHQHHGHGLQWKHVROXWLRQSURFHGXUH 6DIHW\OHDGWLPHVIRUSXUFKDVHGFRPSRQHQWV 7KLV SUREOHP ZDV LQLWLDOO\ IRUPXODWHG LQ +RSS DQG 6SHDUPDQ KHQFHIRUWK +6 &RQVLGHU WKH SUREOHP RI VHWWLQJ VDIHW\ OHDGWLPHV IRU SXUFKDVHGFRPSRQHQWVZKHUHWKHRQO\PDQXIDFWXULQJRSHUDWLRQLVILQDO DVVHPEO\ 7KHUH DUH Q FRPSRQHQWV DQG GHOLYHU\ WLPHV IRU HDFK FRPSRQHQW DUHLQGHSHQGHQWUDQGRPYDULDEOHVZLWK&') IRUFRPSRQHQW L RI )L \ DQG PHDQ µL L Q 7KH RSWLPL]DWLRQ SUREOHP LV WR
CHAPTER 23 INVENTORY ANALYSIS
413
determine the lead-times used to order the components (n decision variable), given relevant costs (that will be detailed shortly). The authors (HS) developed an approximate solution procedure, assuming normality for lead times. An alternative solution procedure, extended for non-normal cases and numerically demonstrated for the normal distribution, is given in Shore (1995a). In the latter reference, the solution procedure is based on the piece-wise linear approximations (Chapter 21). However, as alluded to at the beginning of Section 23.3, these approximations are unbounded in both directions and may deliver unsatisfactory accuracy, for lower-tail quantiles, if the approximated random variable is bounded from below (as lead-time demand is). Additionally, being linear, these approximations fail to deliver satisfactory accuracy if the optimal quantile value resides in the far righttail of the distribution. Here, the family of distributions, introduced in Section 23.3, is employed to derive a general solution for this problem. The accuracy obtained for non-normal populations is demonstrated. Only the first of the two formulations in HS will be addressed, however extension to the other formulation is straightforward. Denote by li the lead-time used to order component i (a decision variable, i=1,2,..,n). Assume that the holding cost per day, for a component that arrives too early, is hi= rpi/365, where r is the annual inventory carrying cost rate, and pi is the cost of component i. It is assumed that assembly cannot be performed early, namely, assembly does not start when the last component has arrived, but according to a pre-planned schedule. However, if at least one component is late, assembly is delayed at a time-independent cost of C. Denote by W(li) the average waiting time for component i: W(li) =
∫
li
y =0
(li - y)dFi(y).
(23.35)
The cost equation is (find further details in HS) Z=
∑
n i =1
hiW(li) + C[1-
Π
n i=1
Pi],
(23.36)
where Pi= Fi(li), namely, Pi is the probability that component i will arrive on time (not be late). It is easy to verify that W(li)= LY(Pi) + li - µ i.
5(63216(02'(/,1*0(7+2'2/2*<
,QWURGXFLQJ LQWR DQG WKHQ IRU /< 3L IURP DQG OL IURP RQHREWDLQVRQGLIIHUHQWLDWLQJZLWKUHVSHFWWR3LDQGHTXDWLQJWR ]HUR $N%N >3L 3L @%N KL &+
L Q N RU
ZKHUH3L )L OL OL LVWKHRSWLPDOOHDGWLPHIRUFRPSRQHQWLDQG
+
Q
h 3 M
M
)URP DQGWKHH[SUHVVLRQIRU+ WKHYDOXHV^3L `PD\EH FRPSXWHG $QXPHULFDOH[DPSOH 6XSSRVH WKDW WKHUH DUH WZR FRPSRQHQWV DQG GHOLYHU\ WLPHV DUH LQGHSHQGHQWO\ DQG LGHQWLFDOO\ JDPPD GLVWULEXWHG ZLWK SDUDPHWHUV α β VHFRQGFDVHLQ7DEOH )XUWKHUDVVXPHWKDW) OL !>N LQ @7KHFRVWGDWDWDNHQIURP&DVHLQ+6DUH S S U 7KHVH \LHOG K K )RU & ZH WDNH & LQVWHDG RI & LQ+6 7KHODWWHU VQXPHULFDOH[DPSOHFRPSULVHVFRPSRQHQWV DQGXVLQJ& IRURXUFDVHZRXOG\LHOGDWULYLDOVROXWLRQRI3 3 ≅ )RUWKHDERYHJDPPDGLVWULEXWLRQ $ % >REWDLQHG IURPDWZRPRPHQWQXPHULFDOILWWLQJRI IRUN @ ,QWURGXFLQJLQWR WKUHHHTXDWLRQVQHHGWREHVROYHG >3 3 @ > 3 @+
IRUWKHILUVWFRPSRQHQW
>3 3 @ > 3 @+
3 3 +
IRUWKHVHFRQGFRPSRQHQW
6ROYLQJWKHVHHTXDWLRQVWKHDSSUR[LPDWHRSWLPDOVROXWLRQLV
&+$37(5,19(1725<$1$/<6,6
3 3
7KLVVROXWLRQ\LHOGVWRWDOFRVWRI= 7KH H[DFW VROXWLRQ VHDUFKHG QXPHULFDOO\ ZLWK WKH H[DFW VWDWLVWLFDO H[SUHVVLRQV LQFOXGLQJUHSHDWHGQXPHULFDOLQWHJUDWLRQ LV 3 3 7KLVVROXWLRQ\LHOGVWRWDOFRVWRI= $ KLJKO\ DFFXUDWH VROXWLRQ LV REWDLQHG %\ GHULYLQJ WKH RSWLPDO VROXWLRQ XVLQJ DQ LQFRUUHFW OHDGWLPH GLVWULEXWLRQ ZKLOVW SUHVHUYLQJ WKH DFWXDOPHDQDQGVWDQGDUGGHYLDWLRQWKHUHDGHUPD\ZLVKWRH[DPLQHWKH VHQVLWLYLW\RIWKLVVROXWLRQDVZDVLPSOHPHQWHGIRUWKHSUHYLRXVH[DPSOH 5HIHUHQFHV >@ +DGOH\ * :KLWLQ 7 $QDO\VLV RI ,QYHQWRU\ 6\VWHPV 3UHQWLFH+DOO (QJOHZRRG&OLIIV1- >@ +RSS : - 6SHDUPDQ 0 / 6HWWLQJ VDIHW\ OHDGWLPHV IRU SXUFKDVHG FRPSRQHQWVLQDVVHPEO\V\VWHPV,,(7UDQVDFWLRQV >@ /DX $ + /DX + 6 $ VLPSOH FRVW PLQLPL]DWLRQ SURFHGXUH IRU WKH 45 LQYHQWRU\PRGHO 'HYHORSPHQWDQGHYDOXDWLRQ,,(7UDQVDFWLRQV >@/DX+6/DX$+ 1RQUREXVWQHVVRIWKHQRUPDODSSUR[LPDWLRQRIOHDG WLPHGHPDQGLQD 45 V\VWHP1DYDO5HVHDUFK/RJLVWLFV >@ 0RRQ , 6LOYHU ( $ 7KH PXOWLLWHP QHZVYHQGRU SUREOHP ZLWK D EXGJHW FRQVWUDLQWDQGIL[HGRUGHULQJFRVWV-RXUQDORIWKH2SHUDWLRQDO5HVHDUFK6RFLHW\ >@3ODWW'(5RELQVRQ/:)UHXQG5% 7UDFWDEOH 45 KHXULVWLFPRGHOV IRUFRQVWUDLQHGVHUYLFHOHYHOV0DQDJHPHQW6FLHQFH >@ 6KRUH + 6LPSOH DSSUR[LPDWLRQV IRU WKH LQYHUVH FXPXODWLYH IXQFWLRQ WKH GHQVLW\IXQFWLRQDQGWKHORVVLQWHJUDORIWKHQRUPDOGLVWULEXWLRQ$SSOLHG6WDWLVWLF >@ 6KRUH + D 6LPSOH JHQHUDO DSSUR[LPDWLRQV IRU D UDQGRP YDULDEOH DQG LWV LQYHUVH GLVWULEXWLRQ IXQFWLRQ EDVHG RQ OLQHDU WUDQVIRUPDWLRQV RI D QRQ VNHZHG YDULDWH6,$0-6FL6WDW&RPSXW > @ 6KRUH + E *HQHUDO DSSUR[LPDWH VROXWLRQV IRU VRPH FRPPRQ LQYHQWRU\ PRGHOV-RXUQDORIWKH2SHUDWLRQDO5HVHDUFK6RFLHW\ >@6KRUH+ D 6HWWLQJVDIHW\OHDGWLPHVIRUSXUFKDVHGFRPSRQHQWVLQDVVHPEO\ V\VWHPVDJHQHUDOVROXWLRQSURFHGXUH,,(7UDQVDFWLRQV >@ 6KRUH + E )LWWLQJ D GLVWULEXWLRQ E\ WKH ILUVW WZR PRPHQWV SDUWLDO DQG FRPSOHWH -RXUQDORI&RPSXWDWLRQDO6WDWLVWLFVDQG'DWD$QDO\VLV >@6KRUH+ 2SWLPXPVFKHGXOHIRUSUHYHQWLYHPDLQWHQDQFH $JHQHUDOVROXWLRQ IRU D SDUWLDOO\ VSHFLILHG WLPHWRIDLOXUH GLVWULEXWLRQ -RXUQDO RI 3URGXFWLRQ DQG 2SHUDWLRQV0DQDJHPHQW
5(63216(02'(/,1*0(7+2'2/2*<
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
5HYLHZ4XHVWLRQV
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
5(63216(02'(/,1*0(7+2'2/2*<
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α, λ, µ2`REWDLQHGDVDUHVXOWRILPSOHPHQWLQJQRQOLQHDUOHDVWVTXDUHV 1//6 :KDW FRQFOXVLRQV GR \RX GHULYH IURP WKLV UHSRUW" )RU
5(9,(:48(67,216
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
5(63216(02'(/,1*0(7+2'2/2*<
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
$XWKRU,QGH[
1RWH ,QGHQWHG QDPHV UHODWH WR DXWKRUV¶ IXOO QDPHV DV WKH\ DSSHDU LQ ³5HIHUHQFHV´VHFWLRQV1RQLQGHQWHGQDPHVUHODWHWRDXWKRUV¶QDPHVLQWKHWH[W IRUDXWKRUVZKRVHQDPHVDSSHDULQD³5HIHUHQFHV´VHFWLRQ
$EUDPRZLW] $EUDPRZLW]0 $NDLNH $NDLNH+ $OEHUW $OEHUW- $OW)% $QGHUVRQ'5 $QGRQ5-/ $QWRLQH $QWRLQH& $UQRX[ $UQRX[) %DJE\ %DJE\5- %DL %DL'6 %DODNULVKQDQ %DODNULVKQDQ1 %DOODUG-/ %HQHGHN %HQHGHN*% %HQQH\DQ %HQQH\DQ-&
%HQVRQ %HQVRQ' %RUURU %RUURU&0 %R[ %R[*(3 %UDQWLQJ' %UDXQHU1 %XUNH5- %XUQKDP %XUQKDP.3 %XUU %XUU,: &DR &DR-+ &KDQJ &KDQJ<6 &KHQ. &KKLNDUD56 &KRL,6 &KRRELQHK &KRRELQHK)
&KRX &KRX<0 &KULVWRSRXORV$ &ODSH\URQ &ODSH\URQ(- &OHPHQWV &OHPHQWV-$ &RKHQ &RKHQ$& &RQGUD &RQGUD/: &RRN &RRN5' &RUQLVK &RUQLVK($ &R['5 'DXEHUW 'DXEHUW7( 'DYLV5' 'HODPSDG\0 'LQJ< 'RGVRQ 'RGVRQ% 'RXVOLQ'5 'UDSHU 'UDSHU15 'XDQH 'XDQH-7 'XGHZLF](- (ZHUV- )DUQXP )DUQXP15 )DUULQJWRQ3$ )LVKHU )LVKHU5$ )LWFKHWW0- )RONV
$87+25,1'(; )RONV-/ )UHXQG5%
*DXGRLQ2 *LDXTXH:) *RHO *RHO$/ *ULPVKDZ *ULPVKDZ6' *XJJHQKHLP *XJJHQKHLP($ +DGOH\ +DGOH\* +DOGDQH +DOGDQH-%6 +DOOLGD\ +DOOLGD\' +DPDGD +DPDGD0 +DPDJXFKL< +LOIHUW\00 +LOO-0 +RDJOLQ +RDJOLQ'& +RSS +RSS:- ,DQQLQR$ ,VODP7 -RKQVRQ -RKQVRQ1/ .DPLQVN\ .DPLQVN\)& .DULDQ .DULDQ=$ .HPS$:
$87+25,1'(; .HQGDOO .HQGDOO0 .LOSDWULFN-( .QDIO .QDIO*- .RFKHUODNRWD6 .RW] .RW]6 /DX /DX$+ /DX$+/ /DX+6 /HZLV /HZLV6/ /L /L% /LQ /LQ/, /LQGEHUJ /LQGEHUJ%& /RYHODFH&5 0DMHU9 0DNQL& 0DOORZ 0DOORZ&3 0DUWLQ% 0DUWLQ-) 0DVRQ5/ 0DUWLQ 0F&XOORFK 0F&XOORFK&( 0F(DFKHUQ'0-U 0HDGH 0HDGH1 0HHNHU 0HHNHU:4 0HVVLPHU6/ 0RQWJRPHU\'& 0RRQ 0RRQ,
0RRUH%' 0RUJDQ- 0RVWHOOHU) 0RWXOVN\ 0RWXOVN\+ 0XVD 0XVD-' 0\HUV 0\HUV5+ 1HOGHU-$ 1HOVRQ35 2HJHUOH 2HJHUOH:5 2NXPRWR. 2UG-. 2VDNL6 2VERUQ 2VERUQ$* 2YHUVWUHHW 2YHUVWUHHW5 3DUNHU' 3DVFXDO 3DVFXDO)* 3HDUVRQ 3HDUVRQ(6 3HQWHUPDQQ: 3ODWW 3ODWW'( 3RODQVN\ 3RODQVN\$0 3RODVHN: 3XWQDP 3XWQDP:(
4XHVHQEHUU\ 4XHVHQEHUU\&3 5DFNHWW 5DFNHWW+* 5DPEHUJ 5DPEHUJ-6 5HVQLFN5 5LHGHO 5LHGHO/ 5REHUW 5REHUW& 5RELQVRQ/: 5XQJHU*& 5X]LFND9-U 5\DQ 5\DQ73 6FKLOOLQJ 6FKLOOLQJ(* 6FKPHLVHU%: 6FKURHGHU 6FKURHGHU0 6FKZHUWPDQ1& 6HDUOH65 6KDFKDP0 6KLPRNDZD 6KLPRNDZD7 6KRUH ± ± ± 6KRUH+ 6LOYHU($
$87+25,1'(; 6PLWK 6PLWK'0 6PLWK+ 6SHDUPDQ0/ 6WDF\ 6WDF\(: 6WHJXQ,$ 6WXDUW 6WXDUW$ 6ZDLQ-- 7DGLNDPDOOD 7DGLNDPDOOD35 7XNH\ 7XNH\-: 9DQ=ZHW 9DQ=ZHW:5 9LOODUV 9LOODUV)0+ 9LQLQJ** 9RLW 9RLW(2 9RQHVK() :DJQHU :DJQHU: :DONHU- :DULFN :DULFN- :HGGHUEXUQ :HGGHUEXUQ5:0 :HLVEHUJ6 :KHHOHU :KHHOHU'- :KLWLQ7 :KLWWHQ%- :LOVRQ :LOVRQ(%
$87+25,1'(; :RRGDOO :RRGDOO:+ :X :X&)- :X+6 ;LH ;LH0
This page intentionally left blank
6XEMHFW,QGH[
$GMXVWHG5VTXDUHG $GRSWLRQUDWH $,& VHH$NDLNH V $OLDVHG $QGHUVRQ'DUOLQJ $QWRLQHHTXDWLRQ $SSUR[LPDWLRQV $UFVLQWUDQVIRUPDWLRQ $5/ VHHDOVRDYHUDJHUXQOHQJWK $UUKHQLXV $VVHPEO\ $WWULEXWHV ±± $YHUDJHUXQOHQJWK VHHDOVR$5/ $YHUDJHZDLWLQJWLPH %DURPHWULFIRUPXOD %& VHHDOVR%R[&R[ ±
%HWD GLVIXQFWLRQ %LJ0DF %LQRPLDO GLV %LYDULDWH GLV %RLOLQJSRLQW %R[&R[ WUDQVIRUPDWLRQVHHDOVR%& %XUU GLV &DQRQLFDO FRUUHODWLRQ DQDO\VLV FRU UHODWLRQYDULDEOH ± &DQRQLFDOOLQN LQ*/0 &DXFK\ GLV &&$ VHH DOVR FDQRQLFDO FRUUHODWLRQ DQDO\VLV ± &HQVRUHGREVHUYDWLRQV &HQWHULQJ &HQWUDO/LPLW7KHRUHP
±
&HQWUDOPRPHQWV &KDUDFWHULVWLFIXQFWLRQ &KDUDFWHUL]DWLRQ &KLVTXDUH GLV &ODSH\URQHTXDWLRQ &OHPHQWV PHWKRG IRU QRQQRUPDO 3&,V &RHIILFLHQWRIYDULDWLRQ &RPSOHWHPRGHO &RPSOHWHPRPHQWV &RPSOLDQFH ± &RQFDYLW\ &RQGLWLRQDO &RQGLWLRQLQJ &RQILGHQFHLQWHUYDOV &RQIOXHQWK\SHUJHRPHWULF IXQFWLRQ &RQIRUPLQJLWHPV &RQWLQXRXVUHYLHZ 45 PRGHO &RQWUROOLPLWV ± ± &RQYHUJHQFHWRQRUPDOLW\ &RQYH[LW\ ± ± &RQYH\RU &RUQLVK)LVKHU H[SDQVLRQV &RUUHODWLRQ
68%-(&7,1'(; &RVWHTXDWLRQ &RYHUDJH>RI 6N.X SODQH@ &ULWLFDOWHPSHUDWXUH &XPXODQWV &XPXODWLYHGHPDQG LQYHQWRU\ 'DWDWUDQVIRUPDWLRQ 'HFRXSOLQJ RIPHDQIURPYDULDQFH 'HJUHHVRIIUHHGRP 'HOLYHU\WLPHV LQYHQWRU\ 'HYLDQFH LQ*/0 'LDJQRVWLF 'LIIXVLRQ SURFHVV ± 'LVFULPLQDQWDQDO\VLV 'LVSHUVLRQSDUDPHWHU LQ*/0 'LVWDQFHPHDVXUHV 'XDOHUURUVWUXFWXUH (FRQRPLVW PDJD]LQH (LJHQYDOXH VHHDOVRFDQRQLFDO (ORQJDWLRQ 7XNH\ VGLVWV (PSLULFDOPRGHOLQJ ± ± (QGSRLQWV RIDGLVWULEXWLRQ ± (TXLYDOHQWIRUPV RI500PRGHO ([SHFWDWLRQ
68%-(&7,1'(; ([SHULPHQWDOGDWD ([SHULPHQWDOGHVLJQ ([SRQHQWLDO GLVIDPLO\ ([WHQGHGORJLVWLF GLV ([WHUQDOO\JHQHUDWHGYDULDWLRQ ([WUHPHYDOXH GLV )WHVW )DFWRULDOPRGHO )DOVHDODUP )DWLJXH OLIHOLIHOLPLW ± )LHOGGDWD )LVKHU VWUDQVIRUPDWLRQ )OR\G GLV )RUHFDVWLQJ ± )RXUPRPHQW PDWFKLQJ ± ± )UDFWLRQDOIDFWRULDO )UDWLR *DOD[\ *DPPD GLV ± *DVKHDWFDSDFLW\ *%' JHQHUDOL]HGEHWDGLV *FKDUW
*HQHUDOL]HGHVWLPDWLQJHTXDWLRQV LQ*/0 *HQHUDOL]HGOLQHDUPRGHOLQJ */0 *HRPHWULF DYHUDJHGLV */' JHQHUDOL]HG /DPEGD GLV VHH DOVR/DPEGD */0 VHH DOVR JHQHUDOL]HG OLQHDU PRGHOLQJ ± ± ± *RPSHUW] PRGHO *RRGQHVVRIILW *UDP&KDUOLHUH[SDQVLRQV *URZWKPRGHOV +DOIQRUPDO GLV +FKDUW FRQWUROFKDUW +HDW ± +HXULVWLF +LHUDUFK\ +RPRJHQHLW\ RIYDULDQFH +RPRVFHGDVWLFLW\ VHH DOVR KRPR JHQHLW\ +\SHUJHRPHWULF GLV ,PDJLQDU\ YDOXHV ,QGLDQD7ZLQ6WXG\ ,QIOHFWLRQV ,QQHULQVWDELOLW\
,17 VHH DOVR LQYHUVH QRUPDOL]LQJ WUDQVIRUPDWLRQV ± ± ± ,QWHUDFWLRQV ,QWHUFHSW ,QWHUSRODWLRQ ,QWUDJDODFWLF YHORFLWLHV ,QYHQWRU\ DQDO\VLVPRGHOV ± ,QYHUVHGLVWULEXWLRQIXQFWLRQ ,QYHUVHIRUHFDVWV ,QYHUVH*DXVVLDQ GLV ,QYHUVHQRUPDOL]LQJWUDQVIRUPDWLRQV ,WHUDWLYHQDWXUHRIHPSLULFDO PRGHOLQJ ,WHUDWLYHO\UHZHLJKWHG OHDVWVTXDUHV -DFRE¶VODGGHU -RKQVRQ IDPLOLHVRIGLVWULEXWLRQV .HUQHOGHQVLW\HVWLPDWLRQ .LQHWLFHQHUJ\ .ROPRJRURY6PLUQRY JRRGQHVVRIILW WHVW .XUWRVLV
68%-(&7,1'(; /DFNRIILW /DGGHU ± /DPEGD GLV /HDGWLPH LQYHQWRU\ ±± /HIWFHQVRUHG /HY\GLVWULEXWLRQ /) /RVV)XQFWLRQ /LQHDUSUHGLFWRU /3 ± ± ± /LQHDUUHJUHVVLRQ ± ± /LQN IXQFWLRQLQ*/0 ±
±
/LTXLGGHQVLW\ /LTXLGKHDWFDSDFLW\ /LTXLGYLVFRVLW\ /RFDWLRQSDUDPHWHU /RJLVWLF GLV /RJLVWLFUHJUHVVLRQ */0 /RJLW OLQNLQ*/0 /RJOLNHOLKRRG
68%-(&7,1'(; /RJOLQN IXQFWLRQLQ*/0 /RJQRUPDO GLV /RJWUDQVIRUPHG VFDOHGDWD ± /RVVIXQFWLRQ VHHDOVR/) ±± /RWVL]H LQYHQWRU\ /3 VHHOLQHDUSUHGLFWRU /SQRUP /7$ ORZHUWDLODUHD 0DLQHIIHFWV 0DOSHUIRUPDQFH 0DQVILHOG PRGHO 0D[LPXPOLNHOLKRRG 0/ 0HDQIXQFWLRQ 0HDQVTXDUHGHUURUV 06(V 0HDVXUHPHQWHUURUV 0HDVXUHPHQWLPSUHFLVLRQ 0HGLDQ ± ± 0HGLDQ5DQN 0HPRU\OHVVSURSHUW\ 0*) VHH DOVR PRPHQW JHQHUDWLQJ IXQFWLRQ 0LQLPL]DWLRQ 0LQLPXPGLVWDQFHPHDVXUHV
0LVVLQJYDOXH 0/ VHHDOVRPD[LPXPOLNHOLKRRG 0RGLILHGQRUPDODSSUR[LPDWLRQV 0RPHQWPDWFKLQJ 0RPHQWJHQHUDWLQJIXQFWLRQ VHHDOVR0*) 0RQRWRQH ± 0RQWH&DUORVLPXODWLRQ 06( VHHDOVRPHDQVTXDUHGHUURU 0XOWLFROOLQHDULW\ 0XOWLSOHOLQHDUUHJUHVVLRQ 0XOWLSOLFDWLYH
0XVD VGDWDVHWV
1HJDWLYHELQRPLDO GLV 1HZWRQ VODZRIFRROLQJ 1//6 VHH DOVR QRQOLQHDU OHDVW VTXDUHV ± ± 1RQFHQWUDOPRPHQWV 1RQKRPRJHQRXV3RLVVRQ 1RQOLQHDUOHDVWVTXDUHV VHHDOVR 1//6 1RQOLQHDUUHJUHVVLRQ 1RQQRUPDOHQYLURQPHQWV
1RUPDOVFHQDULR 1RUPDOLW\DVVXPSWLRQ 1RUPDOL]LQJ WUDQVIRUPDWLRQ 1XFOHDUUHDFWRU 1XPHULFDOLQWHJUDWLRQ 1XPHULFDOVHDUFK 2EMHFWLYHIXQFWLRQ 2IIOLQH 2IIVSULQJ,17V 2UGHUVWDWLVWLFV 2UGHUXSWR 2ULJLQ,17 2XWOLHUV 2YHUGLVSHUVLRQ LQ*/0 3FRQWUROFKDUW 3DUDPHWHUULFK 3DUHWR GLV 3DUVLPRQ\ 3DUWLDO PRPHQWV ± ± 3&$ VHH DOVR SURFHVV FDSDELOLW\ DQDO\VLV 3&, VHH DOVR SURFHVV FDSDELOLW\ LQGLFHV 3HDUVRQ IDPLO\RIGLVWV
68%-(&7,1'(; 3HQDOWLHVIRUVKRUWDJHV 3HUFHQWLOH ± 3HUFHQWLOHPDWFKLQJ 3HUIRUPDQFHPHDVXUHV 3LHFHZLVH OLQHDUDSSUR[LPDWLRQV ± 3ODQFN VUDGLDWLRQODZ 3ODQH 3RLVVRQ 3RO\DW\SHGLVWULEXWLRQV 3RO\QRPLDO 3RZHUODZV 3RZHUVHULHV GLV 3RZHU WUDQVIRUPDWLRQ VHH DOVR %R[ &R[ 3UHGLFWRUV 3UREDELOLW\OLPLWV ± 3URFHVV FDSDELOLW\ DQDO\VLV VHH DOVR 3&$ 3URFHVV FDSDELOLW\ LQGLFHV VHH DOVR 3&, 3URFHVVFRQWUROVFKHPHV 3VHXGRGLVWDQFH 4FKDUWV 4VWDWLVWLF 4XDQWLOH ±±±
68%-(&7,1'(; ±± ± 4XDQWLOHIXQFWLRQ ± ± ± 4XDQWXPQXPEHU 4XDQWXPSK\VLFV 4XDVLOLNHOLKRRG LQ*/0 5DFNHWW¶VHTXDWLRQ 5DGLRDFWLYHGHFD\ 5DQGRPHUURUV 5DQGRP)DWLJXH/LPLW VHHDOVR5)/ 5DQGRPVXP 5D\OHLJK GLV 5HJUHVVRUV 5HODWLYHHUURUV 5HOLDELOLW\ 5HOLDELOLW\JURZWKPRGHOV 5HRUGHUSRLQW LQYHQWRU\ 5HSOHQLVKPHQW LQYHQWRU\ 5HTXLUHPHQWV 5HVLGXDO UHVLGXDO VXP RI VTXDUHV UHVLGXDOVYDULDQFH
5HVSRQVH6XUIDFH0HWKRGRORJ\ 560 5HVSRQVHVXUIDFHV 5)/ VHHDOVR5DQGRP)DWLJXH/LPLW 5LHGHO HTXDWLRQ 5LJKWFHQVRUHG 5REXVWQHVV 5RRW 5RRWPHDQVTXDUHGHUURU 5RRWILQGLQJ 5RRWLGHQWLI\LQJ 560 VHH DOVR UHVSRQVH VXUIDFH PHWKRGRORJ\ 566 UHVLGXDOVXPRIVTXDUHV 6DPSOHFRUUHODWLRQ 6DPSOHVL]HGHWHUPLQDWLRQ 6DPSOLQJHUURU 6DPSOLQJ3UREOHP 6% -RKQVRQIDPLO\ 6FDOHLQYDULDQW 6FDOHSDUDPHWHU 6FDOLQJ 6FRUHIXQFWLRQ LQ*/0 6FXUYH 6HOIJHQHUDWHGYDULDWLRQ 6HY GLV VHH DOVR VPDOOHVW H[WUHPH YDOXH 6KDUHGIHDWXUHV 6KHZKDUW ± ± ± 6LJPRLGFXUYH 6LJQLILFDQFHOHYHO
6NHZQHVV ±± 6NHZQHVVLQIODWLRQ 6PDOOHVWH[WUHPHYDOXH GLV 61FXUYHV 6RIWZDUH 6RIWZDUHUHOLDELOLW\JURZWK 6ROLGGHQVLW\ 6ROLGKHDWFDSDFLW\ 63& VHH DOVR VWDWLVWLFDO SURFHVV FRQWURO ± 6SHFLDOFDVHV 6SHFLDO7KHRU\RI5HODWLYLW\ 6TXDUHURRWOLQN IXQFWLRQLQ*/0 6WDELOL]LQJWKHYDULDQFH 6WDEOHGLVWULEXWLRQV 6WDQGDUGHUURU 67$7,67,&$ 6WDWLVWLFDOFRQWURO 6WDWLVWLFDO SURFHVV FRQWURO VHH DOVR 63& 6WDWLVWLFDOVWDELOLW\ 6WHSZLVH ±
68%-(&7,1'(; 6WRFKDVWLFRSWLPL]DWLRQ 6WUHVV ± 6WUXFWXDOSDUDPHWHUV 6WXGHQW¶VW 6WXGHQWL]HGUHVLGXDOV 68 -RKQVRQIDPLO\ 6XPRIVTXDUHV 6XUIDFHWHQVLRQ 7D\ORU ± 7HFKQRORJLFDOLQQRYDWLRQ 7KHUPDOFRQGXFWLYLW\ 7KHUPLVWRU 7UDQVPLWWHG YDULDWLRQ 7UHQGFXUYHV 7ULSOHSRLQW 7XNH\¶VJDQGKV\VWHPVRIGLVWV 7ZRPRPHQWILWWLQJ 8QVWDEOH 87$ XSSHUWDLODUHD 9DSRUSUHVVXUH ± 9DULDQFHIXQFWLRQ LQ*/0 9DULDWLRQV RIWKH500PRGHO 9LVFRVLW\
68%-(&7,1'(; :DJQHU HTXDWLRQ :HLEXOO GLV :HLJKWHGDYHUDJH :HLJKWHGQRQOLQHDUOHDVWVTXDUHV VHHDOVR:1//6
:HLJKWHGYDULDQFH :HLJKWV :KLWH¶VSORWWLQJSRVLWLRQ :LOVRQDQG+LOIHUW\ DSSUR[LPDWLRQV :1//6 VHH DOVR ZHLJKWHG QRQ OLQHDUOHDVWVTXDUHV
